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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">P</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> holds   <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Olive" title="b2">P</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> holds   <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Olive" title="b1">P</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">x</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">P</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:12"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c6">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:12"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E3:12"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="weierstr.html#E3:12"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A1</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A2</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">A1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E8:12"/>
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E4:12"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:12"/><b>then </b><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E10:12"/>
<font color="Maroon" title="c8">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E3:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="weierstr.html#D2" target="_self" title="WEIERSTR:def.2">Def2</a></i>;<br/>
<a NAME="E11:12"/><b>then </b>
<font color="Maroon" title="c9">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="weierstr.html#D1" target="_self" title="WEIERSTR:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E12:12"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:12"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="weierstr.html#E9:12"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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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