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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font> being   <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b1">c</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b2">r1</font>,<font color="Olive" title="b3">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is   <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font> be   <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c2">c</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is   <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is   <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c2">c</font> as   <a href="topalg_6.html#V4" title="TOPALG_6:attr.4">parametrized-curve</a>  <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T23" target="_self" onmouseover="rs('topalg_6/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f1</font> = <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">A</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> as   <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#D4" target="_self" onmouseover="rs('topalg_6/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E4:55"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B</font> is   <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 
;<br/>
<a NAME="E5:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> is   <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">S</font> being    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> , <font color="Maroon" title="c10">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c9">S</font>,<font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E7:55"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Maroon" title="c9">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c10">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#D4" target="_self" onmouseover="rs('topalg_6/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">A0</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ;<br/>
<a NAME="E9:55"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c9">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">K0</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:55"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">g1</font> = <font color="Maroon" title="c10">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c12">K0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">K0</font></span>)</span>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T9" onmouseover="rs('pre_topc/T9')" onmouseout="rh()">PRE_TOPC:9</a></span>;<br/>
<a NAME="E12:55"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c13">g1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="topmetr.html#T7" onmouseover="rs('topmetr/T7')" onmouseout="rh()">TOPMETR:7</a></span>;<br/>
<span class="lab"><font color="Green" title="E35">A6</font></span>: <font color="Maroon" title="c13">g1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="relat_1.html#T71" onmouseover="rs('relat_1/T71')" onmouseout="rh()">RELAT_1:71</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>

;<br/>
<a NAME="E14:55"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">K1</font> = <font color="Maroon" title="c12">K0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">A0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">D1</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a>, <a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<a NAME="E17:55"/>
<font color="Maroon" title="c9">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">K0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E14:55"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="pre_topc.html#T7" onmouseover="rs('pre_topc/T7')" onmouseout="rh()">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">f1</font> = <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="topalg_6.html#V4" title="TOPALG_6:attr.4">parametrized-curve</a>  <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="topalg_6.html#D4" target="_self" onmouseover="rs('topalg_6/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E19:55"/>
<font color="Maroon" title="c2">c</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f1</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E20:55"/>
<font color="Maroon" title="c2">c</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r1</font>,<font color="Maroon" title="c4">r2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is   <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T20" target="_self" onmouseover="rs('topalg_6/T20')" onmouseout="rh()">Th20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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