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<b>let </b><font color="Maroon" title="c10">p1</font>, <font color="Maroon" title="c11">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <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">P</font>, <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">P2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c10">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">p2</font> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font> &amp; <font color="Olive" title="b3">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">P2</font> &amp; <font color="Olive" title="b2">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Olive" title="b1">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a></span><br/><b>let </b><font color="Maroon" title="c12">P</font>, <font color="Maroon" title="c13">P1</font>, <font color="Maroon" title="c14">P2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">p2</font> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c13">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font> &amp; <font color="Maroon" title="c12">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">P2</font> &amp; <font color="Maroon" title="c13">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <font color="Maroon" title="c12">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:14"/>
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">p2</font> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">P</font> )
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c13">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font>
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c14">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font>
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c12">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">P2</font>
 <b>and </b><br/><a NAME="E5:14"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c13">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a></span><br/>

<b>consider </b><font color="Maroon" title="c15">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">P1</font></span>)</span><b> such that </b><br/><a NAME="E7:14"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c15">f1</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>and </b><br/><a NAME="E8:14"/><i><font color="Green" title="E38">A6</font></i>: 
( <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> &amp; <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> )
 <b>by </b><i><a class="txt" href="#E2:14"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">f2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c14">P2</font></span>)</span><b> such that </b><br/><a NAME="E10:14"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c16">f2</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>and </b><br/><a NAME="E11:14"/><i><font color="Green" title="E40">A8</font></i>: 
( <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> )
 <b>by </b><i><a class="txt" href="#E3:14"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">h1</font>, <font color="Maroon" title="c18">h2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:14"/><i><font color="Green" title="E41">A9</font></i>: 
( <font color="Maroon" title="c17">h1</font> is <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> &amp; <font color="Maroon" title="c18">h2</font> is <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> )
 <b>and </b><br/><a NAME="E14:14"/><i><font color="Green" title="E42">A10</font></i>: 
(  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>and </b><br/><a NAME="E15:14"/><i><font color="Green" title="E43">A11</font></i>: 
( <font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>and </b><br/><a NAME="E16:14"/><i><font color="Green" title="E44">A12</font></i>: 
( <font color="Maroon" title="c18">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c18">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="topreal1.html#T30" title="TOPREAL1:th.30">TOPREAL1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">L1</font> =  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">h1</font>, <font color="Maroon" title="c20">L2</font> =  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E14:14"><i><font color="Green" title="E42">A10</font></i></a></i>;<br/>
<a NAME="E18:14"/>
<font color="Maroon" title="c19">L1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="#E13:14"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="#E15:14"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="topreal1.html#T31" title="TOPREAL1:th.31">TOPREAL1:31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">g1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">L1</font></span>)</span><b> such that </b><br/><a NAME="E20:14"/><i><font color="Green" title="E45">A13</font></i>: 
( <font color="Maroon" title="c21">g1</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> &amp; <font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E21:14"/>
<font color="Maroon" title="c20">L2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="#E13:14"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="#E16:14"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="topreal1.html#T31" title="TOPREAL1:th.31">TOPREAL1:31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">g2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">L2</font></span>)</span><b> such that </b><br/><a NAME="E23:14"/><i><font color="Green" title="E46">A14</font></i>: 
( <font color="Maroon" title="c22">g2</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> &amp; <font color="Maroon" title="c22">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c22">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E24:14"/><i><font color="Green" title="E47">A15</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P1</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c13">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">P</font> )
 
<b>by </b><i><a class="txt" href="#E4:14"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">P'</font> = <font color="Maroon" title="c12">P</font>, <font color="Maroon" title="c24">P1'</font> = <font color="Maroon" title="c13">P1</font>, <font color="Maroon" title="c25">P2'</font> = <font color="Maroon" title="c14">P2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E26:14"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">f1</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E24:14"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">ff1</font> = <font color="Maroon" title="c15">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c23">P'</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E28:14"/>
( <font color="Maroon" title="c15">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">P1'</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c23">P'</font> )
 
<b>by </b><i><a class="txt" href="#E7:14"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="#E24:14"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E29:14"/><b>then </b><i><font color="Green" title="E48">A16</font></i>: 
<font color="Maroon" title="c26">ff1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></i>;<br/>
<a NAME="E30:14"/><i><font color="Green" title="E49">A17</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c14">P2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P2</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">P</font> &amp; <font color="Maroon" title="c14">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">P</font> )
 
<b>by </b><i><a class="txt" href="#E4:14"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E31:14"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">f2</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c27">ff2</font> = <font color="Maroon" title="c16">f2</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c23">P'</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E33:14"/>
( <font color="Maroon" title="c16">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">P2'</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c23">P'</font> )
 
<b>by </b><i><a class="txt" href="#E10:14"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="#E30:14"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E34:14"/><b>then </b><i><font color="Green" title="E50">A18</font></i>: 
<font color="Maroon" title="c27">ff2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></i>;<br/>
<b>set </b><font color="Maroon" title="c28">k1</font> = <font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c29">k2</font> = <font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">RS</font> =  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E36:14"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">h2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 
<b>by </b><i><a class="txt" href="#E13:14"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c31">Lh1</font> =  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">h1</font>, <font color="Maroon" title="c32">Lh2</font> =  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T29" title="TOPREAL1:th.29">TOPREAL1:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">g1</font> = <font color="Maroon" title="c21">g1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c34">g2</font> = <font color="Maroon" title="c22">g2</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c35">T1</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font>;<br/>
<b>set </b><font color="Maroon" title="c36">T2</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font>;<br/>
<b>set </b><font color="Maroon" title="c37">T</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font>;<br/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font></span>)</span>

;<br/>
<b>then reconsider </b><font color="Maroon" title="c38">p00</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <font color="Maroon" title="c39">p11</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font></span>)</span> <b>by </b><i><a class="txt" href="topreal2.html#E21"><i><font color="Green" title="E21">Lm21</font></i></a>, <a class="txt" href="topreal2.html#E22"><i><font color="Green" title="E22">Lm22</font></i></a>, <a class="ref" href="topreal1.html#T20" title="TOPREAL1:th.20">TOPREAL1:20</a></i>;<br/>
<a NAME="E42:14"/><i><font color="Green" title="E51">A19</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c17">h1</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c18">h2</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E43:14"/><i><font color="Green" title="E52">A20</font></i>: 
 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11" title="HEINE:th.11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E44:14"/><i><font color="Green" title="E53">A21</font></i>: 
( <font color="Maroon" title="c33">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c34">g2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c33">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c34">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c34">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E45:14"/><b>then </b><i><font color="Green" title="E54">A22</font></i>: 
( <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> )
 
<b>by </b><i><a class="txt" href="#E43:14"><i><font color="Green" title="E52">A20</font></i></a>, <a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></i>;<br/>
<a NAME="E46:14"/><i><font color="Green" title="E55">A23</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font> is <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T3" title="TOPMETR:th.3">TOPMETR:3</a></i>;<br/>
<a NAME="E47:14"/>
( <font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E48:14"/><b>then </b><i><font color="Green" title="E56">A24</font></i>: 
( <font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="#E29:14"><i><font color="Green" title="E48">A16</font></i></a>, <a class="txt" href="#E34:14"><i><font color="Green" title="E50">A18</font></i></a></i>;<br/>
<a NAME="E49:14"/><i><font color="Green" title="E57">A25</font></i>: 
( <font color="Maroon" title="c33">g1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c34">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E50:14"/><b>then </b><i><font color="Green" title="E58">A26</font></i>: 
<font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">g2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="#E44:14"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E51:14"/><i><font color="Green" title="E59">A27</font></i>: 
<font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">g1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="#E44:14"><i><font color="Green" title="E53">A21</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E52:14"/><i><font color="Green" title="E60">A28</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c34">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">ff1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">ff2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E53:14"/><b>then </b><i><font color="Green" title="E61">A29</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">g1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c34">g2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E54:14"/><i><font color="Green" title="E62">A30</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">ff2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">ff1</font> )
 
<b>by </b><i><a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E55:14"/><i><font color="Green" title="E63">A31</font></i>: 
<font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c34">g2</font>
 
<b>by </b><i><a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E53:14"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E56:14"/><i><font color="Green" title="E64">A32</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c34">g2</font>
 
<b>by </b><i><a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E57:14"/><i><font color="Green" title="E65">A33</font></i>: 
<font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E55:14"><i><font color="Green" title="E63">A31</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E58:14"/><i><font color="Green" title="E66">A34</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font>
 
<b>by </b><i><a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E53:14"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E59:14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">ff2</font>
 
<b>by </b><i><a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E53:14"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E60:14"/><b>then </b><i><font color="Green" title="E67">A35</font></i>: 
<font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E57:14"><i><font color="Green" title="E65">A33</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E61:14"/><i><font color="Green" title="E68">A36</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c33">g1</font>
 
<b>by </b><i><a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E62:14"/><b>then </b><i><font color="Green" title="E69">A37</font></i>: 
<font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E53:14"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E63:14"/><i><font color="Green" title="E70">A38</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font>
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="#E53:14"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E64:14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">ff1</font>
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E53:14"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E65:14"/><b>then </b>
<font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E62:14"><i><font color="Green" title="E69">A37</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>then </b><i><font color="Green" title="E71">A39</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font> = 
<font color="Maroon" title="c26">ff1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c10">p1</font>
<b>by </b><i><a class="txt" href="#E8:14"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="#E53:14"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E72">A40</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font> = 
<font color="Maroon" title="c27">ff2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:14"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E53:14"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">p00</font>
<b>by </b><i><a class="txt" href="#E60:14"><i><font color="Green" title="E67">A35</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E68:14"/><i><font color="Green" title="E73">A41</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">g1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c34">g2</font> )
 
<b>by </b><i><a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E69:14"/><i><font color="Green" title="E74">A42</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">ff2</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">ff1</font> )
 
<b>by </b><i><a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E70:14"/><i><font color="Green" title="E75">A43</font></i>: 
( <font color="Maroon" title="c39">p11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> &amp; <font color="Maroon" title="c39">p11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E56:14"><i><font color="Green" title="E64">A32</font></i></a>, <a class="txt" href="#E61:14"><i><font color="Green" title="E68">A36</font></i></a>, <a class="txt" href="#E68:14"><i><font color="Green" title="E73">A41</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E71:14"/><i><font color="Green" title="E76">A44</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font>
 
<b>by </b><i><a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E68:14"><i><font color="Green" title="E73">A41</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E72:14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">ff2</font>
 
<b>by </b><i><a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E68:14"><i><font color="Green" title="E73">A41</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E73:14"/><b>then </b><i><font color="Green" title="E77">A45</font></i>: 
<font color="Maroon" title="c39">p11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E70:14"><i><font color="Green" title="E75">A43</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E74:14"/><i><font color="Green" title="E78">A46</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font>
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="#E68:14"><i><font color="Green" title="E73">A41</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E75:14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">ff1</font>
 
<b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E68:14"><i><font color="Green" title="E73">A41</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E76:14"/><b>then </b>
<font color="Maroon" title="c39">p11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E70:14"><i><font color="Green" title="E75">A43</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>then </b><i><font color="Green" title="E79">A47</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font> = 
<font color="Maroon" title="c26">ff1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c11">p2</font>
<b>by </b><i><a class="txt" href="#E8:14"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="#E68:14"><i><font color="Green" title="E73">A41</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E80">A48</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font> = 
<font color="Maroon" title="c27">ff2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:14"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E68:14"><i><font color="Green" title="E73">A41</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">p11</font>
<b>by </b><i><a class="txt" href="#E73:14"><i><font color="Green" title="E77">A45</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E79:14"/>
( <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font> )
 
<b>by </b><i><a class="txt" href="#E14:14"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="#E42:14"><i><font color="Green" title="E51">A19</font></i></a>, <a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c40">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P</font></span>)</span> <b>by </b><i><a class="txt" href="#E14:14"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="#E42:14"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="#E45:14"><i><font color="Green" title="E54">A22</font></i></a>, <a class="txt" href="#E46:14"><i><font color="Green" title="E55">A23</font></i></a>, <a class="txt" href="#E48:14"><i><font color="Green" title="E56">A24</font></i></a>, <a class="txt" href="#E67:14"><i><font color="Green" title="E72">A40</font></i></a>, <a class="txt" href="#E78:14"><i><font color="Green" title="E80">A48</font></i></a>, <a class="ref" href="compts_1.html#T30" title="COMPTS_1:th.30">COMPTS_1:30</a></i>;<br/>
<a NAME="E81:14"/><i><font color="Green" title="E81">A49</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>
 
<b>by </b><i><a class="ref" href="topreal2.html#T2" target="_self" title="TOPREAL2:th.2">Th2</a>, <a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></i>;<br/>
<a NAME="E82:14"/><i><font color="Green" title="E82">A50</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c23">P'</font> is <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T3" title="TOPMETR:th.3">TOPMETR:3</a></i>;<br/>
<a NAME="E83:14"/><i><font color="Green" title="E83">A51</font></i>: 
<font color="Maroon" title="c15">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E7:14"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E84:14"/><i><font color="Green" title="E84">A52</font></i>: 
<font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a></i>;<br/>
<a NAME="E85:14"/><b>then </b><i><font color="Green" title="E85">A53</font></i>: 
<font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E83:14"><i><font color="Green" title="E83">A51</font></i></a></i>;<br/>
<a NAME="E86:14"/><i><font color="Green" title="E86">A54</font></i>: 
<font color="Maroon" title="c16">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E10:14"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E87:14"/><i><font color="Green" title="E87">A55</font></i>: 
<font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a></i>;<br/>
<a NAME="E88:14"/><b>then </b><i><font color="Green" title="E88">A56</font></i>: 
<font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E86:14"><i><font color="Green" title="E86">A54</font></i></a></i>;<br/>
<a NAME="E89:14"/><i><font color="Green" title="E89">A57</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">g1</font>
 
<b>by </b><i><a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E51:14"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<a NAME="E90:14"/><i><font color="Green" title="E90">A58</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c34">g2</font>
 
<b>by </b><i><a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<a NAME="E91:14"/><b>then </b><i><font color="Green" title="E91">A59</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E92:14"/><i><font color="Green" title="E92">A60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c41">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:14_6"/>
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span><br/>

<a NAME="E2:14_6"/><b>then </b><i><font color="Green" title="E93">A61</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></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#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c41">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E3:14_6"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">Lh1</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Lh2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:14_6"/><b>then </b>
( <font color="Maroon" title="c41">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> or <font color="Maroon" title="c41">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> )
 
<b>by </b><i><a class="txt" href="#E14:14"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="#E42:14"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="#E66:14"><i><font color="Green" title="E71">A39</font></i></a>, <a class="txt" href="#E77:14"><i><font color="Green" title="E79">A47</font></i></a>, <a class="txt" href="#E2:14_6"><i><font color="Green" title="E93">A61</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:14_6"/>
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E57:14"><i><font color="Green" title="E65">A33</font></i></a>, <a class="txt" href="#E66:14"><i><font color="Green" title="E71">A39</font></i></a>, <a class="txt" href="#E67:14"><i><font color="Green" title="E72">A40</font></i></a>, <a class="txt" href="#E70:14"><i><font color="Green" title="E75">A43</font></i></a>, <a class="txt" href="#E77:14"><i><font color="Green" title="E79">A47</font></i></a>, <a class="txt" href="#E78:14"><i><font color="Green" title="E80">A48</font></i></a>, <a class="txt" href="#E91:14"><i><font color="Green" title="E91">A59</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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>
<a NAME="E93:14"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E93">A62</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">f1</font>
<b>by </b><i><a class="txt" href="#E89:14"><i><font color="Green" title="E89">A57</font></i></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c13">P1</font>
<b>by </b><i><a class="txt" href="#E7:14"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="#E24:14"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
;<br/>
<i><font color="Green" title="E94">A63</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">f2</font>
<b>by </b><i><a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E90:14"><i><font color="Green" title="E90">A58</font></i></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c14">P2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E10:14"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c14">P2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E96:14"/><b>then </b><i><font color="Green" title="E95">A64</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c40">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c23">P'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:14"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="#E24:14"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="#E92:14"><i><font color="Green" title="E92">A60</font></i></a>, <a class="txt" href="#E94:14"><i><font color="Green" title="E93">A62</font></i></a>, <a class="ref" href="topmetr2.html#T3" title="TOPMETR2:th.3">TOPMETR2:3</a></i>;<br/>
<a NAME="E97:14"/><i><font color="Green" title="E96">A65</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c40">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c30">RS</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c41">x1</font>, <font color="Maroon" title="c42">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c42">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span> implies  not <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:14_9"/><i><font color="Green" title="E97">A66</font></i>: 
<font color="Maroon" title="c41">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:14_9"/><i><font color="Green" title="E98">A67</font></i>: 
<font color="Maroon" title="c42">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font></span><br/><b>assume </b><a NAME="E3:14_9"/><i><font color="Green" title="E99">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E4:14_9"/><i><font color="Green" title="E100">A69</font></i>: 
<font color="Maroon" title="c42">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">ff1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:14_9"><i><font color="Green" title="E98">A67</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E5:14_9"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P1</font> )
 <b>by </b><i><a class="txt" href="#E94:14"><i><font color="Green" title="E93">A62</font></i></a>, <a class="txt" href="#E95:14"><i><font color="Green" title="E94">A63</font></i></a>, <a class="txt" href="#E1:14_9"><i><font color="Green" title="E97">A66</font></i></a>, <a class="txt" href="#E3:14_9"><i><font color="Green" title="E99">A68</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:14_9"/><b>then </b><i><font color="Green" title="E101">A70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">P2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:14_9"/><i><font color="Green" title="E102">A71</font></i>: 
<font color="Maroon" title="c41">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E1:14_9"><i><font color="Green" title="E97">A66</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E8:14_9"/><i><font color="Green" title="E103">A72</font></i>: 
<font color="Maroon" title="c42">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E4:14_9"><i><font color="Green" title="E100">A69</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_9_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> )
 </span><b>by </b><i><a class="txt" href="#E5:14"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E6:14_9"><i><font color="Green" title="E101">A70</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_9_1_1"><b>suppose </b></a><a NAME="E1:14_9_1_1"/><i><font color="Green" title="E104">A73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:14_9_1_1"/><b>then </b>
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">ff2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">ff2</font> &amp; <font color="Maroon" title="c27">ff2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> )
 <b>by </b><i><a class="txt" href="#E11:14"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="#E1:14_9"><i><font color="Green" title="E97">A66</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E3:14_9_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E49:14"><i><font color="Green" title="E57">A25</font></i></a>, <a class="txt" href="#E50:14"><i><font color="Green" title="E58">A26</font></i></a>, <a class="txt" href="#E54:14"><i><font color="Green" title="E62">A30</font></i></a>, <a class="txt" href="#E55:14"><i><font color="Green" title="E63">A31</font></i></a>, <a class="txt" href="#E86:14"><i><font color="Green" title="E86">A54</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:14_9_1_1"/><b>then </b><i><font color="Green" title="E105">A74</font></i>: 
<font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">p00</font>
 <b>by </b><i><a class="txt" href="#E58:14"><i><font color="Green" title="E66">A34</font></i></a>, <a class="txt" href="#E87:14"><i><font color="Green" title="E87">A55</font></i></a>, <a class="txt" href="#E7:14_9"><i><font color="Green" title="E102">A71</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E5:14_9_1_1"/>
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">ff1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">ff1</font> &amp; <font color="Maroon" title="c26">ff1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> )
 <b>by </b><i><a class="txt" href="#E8:14"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="#E3:14_9"><i><font color="Green" title="E99">A68</font></i></a>, <a class="txt" href="#E4:14_9"><i><font color="Green" title="E100">A69</font></i></a>, <a class="txt" href="#E1:14_9_1_1"><i><font color="Green" title="E104">A73</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E6:14_9_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c38">p00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E54:14"><i><font color="Green" title="E62">A30</font></i></a>, <a class="txt" href="#E61:14"><i><font color="Green" title="E68">A36</font></i></a>, <a class="txt" href="#E83:14"><i><font color="Green" title="E83">A51</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E7:14_9_1_1"/><b>then </b>
<font color="Maroon" title="c42">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E63:14"><i><font color="Green" title="E70">A38</font></i></a>, <a class="txt" href="#E84:14"><i><font color="Green" title="E84">A52</font></i></a>, <a class="txt" href="#E1:14_9"><i><font color="Green" title="E97">A66</font></i></a>, <a class="txt" href="#E8:14_9"><i><font color="Green" title="E103">A72</font></i></a>, <a class="txt" href="#E4:14_9_1_1"><i><font color="Green" title="E105">A74</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E8:14_9_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14_9"><i><font color="Green" title="E98">A67</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_9_1_2"><b>suppose </b></a><a NAME="E1:14_9_1_2"/><i><font color="Green" title="E104">A75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:14_9_1_2"/><b>then </b>
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">ff2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">ff2</font> &amp; <font color="Maroon" title="c27">ff2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> )
 <b>by </b><i><a class="txt" href="#E11:14"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="#E1:14_9"><i><font color="Green" title="E97">A66</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E3:14_9_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c39">p11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E23:14"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E56:14"><i><font color="Green" title="E64">A32</font></i></a>, <a class="txt" href="#E69:14"><i><font color="Green" title="E74">A42</font></i></a>, <a class="txt" href="#E86:14"><i><font color="Green" title="E86">A54</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:14_9_1_2"/><b>then </b><i><font color="Green" title="E105">A76</font></i>: 
<font color="Maroon" title="c41">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">p11</font>
 <b>by </b><i><a class="txt" href="#E71:14"><i><font color="Green" title="E76">A44</font></i></a>, <a class="txt" href="#E87:14"><i><font color="Green" title="E87">A55</font></i></a>, <a class="txt" href="#E7:14_9"><i><font color="Green" title="E102">A71</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E5:14_9_1_2"/>
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">ff1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">ff1</font> &amp; <font color="Maroon" title="c26">ff1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> )
 <b>by </b><i><a class="txt" href="#E8:14"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="#E3:14_9"><i><font color="Green" title="E99">A68</font></i></a>, <a class="txt" href="#E4:14_9"><i><font color="Green" title="E100">A69</font></i></a>, <a class="txt" href="#E1:14_9_1_2"><i><font color="Green" title="E104">A75</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E6:14_9_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c39">p11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">g1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E20:14"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="#E52:14"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E61:14"><i><font color="Green" title="E68">A36</font></i></a>, <a class="txt" href="#E69:14"><i><font color="Green" title="E74">A42</font></i></a>, <a class="txt" href="#E83:14"><i><font color="Green" title="E83">A51</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E7:14_9_1_2"/><b>then </b>
<font color="Maroon" title="c42">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ff2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">g2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E74:14"><i><font color="Green" title="E78">A46</font></i></a>, <a class="txt" href="#E84:14"><i><font color="Green" title="E84">A52</font></i></a>, <a class="txt" href="#E1:14_9"><i><font color="Green" title="E97">A66</font></i></a>, <a class="txt" href="#E8:14_9"><i><font color="Green" title="E103">A72</font></i></a>, <a class="txt" href="#E4:14_9_1_2"><i><font color="Green" title="E105">A76</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E8:14_9_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:14_9"><i><font color="Green" title="E98">A67</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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><b>end;</b></div></div><b>end;</b></div>
<a NAME="E99:14"/><b>then </b><i><font color="Green" title="E97">A77</font></i>: 
<font color="Maroon" title="c40">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="#E85:14"><i><font color="Green" title="E85">A53</font></i></a>, <a class="txt" href="#E88:14"><i><font color="Green" title="E88">A56</font></i></a>, <a class="ref" href="topmetr2.html#T2" title="TOPMETR2:th.2">TOPMETR2:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c41">h</font> = <font color="Maroon" title="c40">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">P</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c41">h</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal2.html#D1" target="_self" title="TOPREAL2:def.1">TOPREAL2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c41">h</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a></span><br/>

<b>thus </b><a NAME="E101:14"/>
<font color="Maroon" title="c41">h</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>by </b><i><a class="txt" href="#E81:14"><i><font color="Green" title="E81">A49</font></i></a>, <a class="txt" href="#E82:14"><i><font color="Green" title="E82">A50</font></i></a>, <a class="txt" href="#E96:14"><i><font color="Green" title="E95">A64</font></i></a>, <a class="txt" href="#E97:14"><i><font color="Green" title="E96">A65</font></i></a>, <a class="txt" href="#E99:14"><i><font color="Green" title="E97">A77</font></i></a>, <a class="ref" href="compts_1.html#T26" title="COMPTS_1:th.26">COMPTS_1:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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