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

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">U</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

















<b>set </b><font color="Maroon">D</font> =  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:106"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">A</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E2:106"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E3:106"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E4:106"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>and </b><br/><a NAME="E5:106"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:106"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E7:106"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E8:106"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E9:106"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E10:106"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">U</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E11:106"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan.html#T12" target="_self">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T2B1</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B1</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2 ;<br/>
<a NAME="E14:106"/><i><font color="Green">E115</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E15:106"/><i><font color="Green">E117</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E16:106"/><i><font color="Green">E118</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3">BROUWER:3</a></i>;<br/>
<a NAME="E17:106"/><i><font color="Green">E119</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T16">TOPREAL9:16</a></i>;<br/>
<a NAME="E18:106"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E6:106"><i><font color="Green">E80</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E6:106"><i><font color="Green">E80</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E20:106"/><i><font color="Green">E121</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan.html#E9:106"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E21:106"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/>
<a NAME="E22:106"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="topreal9.html#T8">TOPREAL9:8</a></i>;<br/>
<a NAME="E23:106"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F1</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <b>by </b><i>, <a class="txt" href="jordan.html#E9:106"><i><font color="Green">E83</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E25:106"/><i><font color="Green">E177</font></i>: 
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17">TOPREAL9:17</a></i>;<br/>
<a NAME="E26:106"/><i><font color="Green">E178</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T19">TOPREAL9:19</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E28:106"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E29:106"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="jordan.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E30:106"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F3</font> = <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <b>by </b><i>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T1</font> = <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">F1</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> ;<br/>
<b>reconsider </b><font color="Maroon">T3</font> = <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">F3</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> ;<br/>
<a NAME="E34:106"/><i><font color="Green">E181</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E35:106"/><i><font color="Green">E182</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F3</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E36:106"/><i><font color="Green">E183</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E37:106"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:106_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:106_1"/><b>then </b><i><font color="Green">E185</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E7:106"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E3:106_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:106"><i><font color="Green">E82</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:106_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="jordan.html#T12" target="_self">Th12</a>, , , , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:106"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">F3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:106_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F3</font>
 ;<br/>

<a NAME="E2:106_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:106_2"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:106_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="txt" href="jordan.html#E9:106"><i><font color="Green">E83</font></i></a>, <a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E39:106"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">F1</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F1</font>,<font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T14" target="_self">Th14</a>, , <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">F1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2B1</font> <b>by </b><i>, <a class="ref" href="jordan.html#T17" target="_self">Th17</a>, , <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">F3</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T3</font>,<font color="Maroon">T2B1</font> <b>by </b><i><a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="jordan.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E42:106"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">F3</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T3</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E43:106"/><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:106_3"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T3</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:106_3"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_3_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_3_1_1"><b>suppose </b></a><a NAME="E1:106_3_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:106_3_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 <b>by </b><i>, <a class="txt" href="jordan.html#E10:106"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:106_3_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T3</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_3_1_2"><b>suppose </b></a><a NAME="E1:106_3_1_2"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:106_3_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E10:106"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><a NAME="E3:106_3_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F3</font>
 <b>by </b><i><a class="txt" href="jordan.html#E10:106"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:106_3_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T3</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">DT</font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> as  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#D14">BORSUK_1:def 14</a>, <a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E45:106"/>
<font color="Maroon">DT</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E46:106"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">F1</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/>
<a NAME="E47:106"/>
<font color="Maroon">P</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_4"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">U</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a><br/>

<b>thus </b><a NAME="E1:106_4"/>
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">U</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="jordan.html#T11" target="_self">Th11</a>, <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:106"/><b>then </b>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T18">SPRECT_3:18</a></i>;<br/>
<a NAME="E49:106"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E50:106"/><b>then </b>
<font color="Maroon">DT</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E51:106"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">F3</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">F1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A1</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E53:106"/><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T2B1</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<a NAME="E54:106"/>
<font color="Maroon">T3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T7">TSEP_1:7</a></i>;<br/>
<a NAME="E55:106"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">T3</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T2B1</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="jordan.html#T17" target="_self">Th17</a>, , <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<a NAME="E56:106"/>
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E57:106"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jordan.html#T21" target="_self">Th21</a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g2</font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">F3</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T3</font>,<font color="Maroon">T3</font> <b>by </b><i><a class="ref" href="jordan.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E59:106"/>
<font color="Maroon">g2</font> <a href="funct_2.html#R2">=</a>  <a href="topgrp_1.html#K4">id</a> <font color="Maroon">T3</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E60:106"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jordan.html#T22" target="_self">Th22</a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E61:106"/><i><font color="Green">E252</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:106_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  )
 ;<br/>

<a NAME="E2:106_5"/><b>then </b><i><font color="Green">E253</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:106_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:106_5"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E62:106"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T3</font></span>)</span> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:106_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T3</font></span>)</span>
 ;<br/>

<a NAME="E2:106_6"/><b>then </b><i><font color="Green">E254</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T3</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:106_6"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:106_6"/><b>then </b><i><font color="Green">E257</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E14:106"><i><font color="Green">E115</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, , <a class="txt" href="jordan.html#E15:106"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span><b> such that </b><br/><a NAME="E64:106"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g1</font>
 <b>and </b><br/><a NAME="E65:106"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, <a class="txt" href="jordan.html#E10:106"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan.html#E11:106"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan.html#T23" target="_self">Th23</a>, , <a class="ref" href="jgraph_2.html#T9">JGRAPH_2:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E66:106"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan.html#E15:106"><i><font color="Green">E117</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>;<br/>

<a NAME="E67:106"/><i><font color="Green">E260</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T3</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:106_7"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:106_7"/><b>then </b><i><font color="Green">E262</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_7_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_7_1_1"><b>suppose </b></a><a NAME="E1:106_7_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E18:106"><i><font color="Green">E120</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_7_1_2"><b>suppose </b></a><a NAME="E1:106_7_1_2"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 ;<br/><div class="add"><a NAME="E2:106_7_1_2"/><b>then </b><i><font color="Green">E264</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="txt" href="jordan.html#E17:106"><i><font color="Green">E119</font></i></a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E20:106"><i><font color="Green">E121</font></i></a>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E20:106"><i><font color="Green">E121</font></i></a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E14:106"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan.html#E22:106"><i><font color="Green">E122</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E69:106"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E70:106"/><b>then </b><i><font color="Green">E265</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F3</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E16:106"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan.html#E17:106"><i><font color="Green">E119</font></i></a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="jordan.html#E17:106"><i><font color="Green">E119</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


</div>
