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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">f1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">X1</font>,<font color="Maroon" title="c2">Y</font><br/> for <font color="Olive" title="b4">f2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">X2</font>,<font color="Maroon" title="c2">Y</font>  st ( <font color="Olive" title="b1">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Olive" title="b2">X2</font> or <font color="Olive" title="b3">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b2">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b2">X2</font></span>)</span> ) holds <br/>( ( for <font color="Olive" title="b5">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">X1</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b4">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b5">A</font> ) &amp; ( for <font color="Olive" title="b5">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">X2</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b4">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b5">A</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font><br/> for <font color="Olive" title="b2">f2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font>  st ( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> or <font color="Olive" title="b1">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> ) holds <br/>( ( for <font color="Olive" title="b3">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">A</font> ) &amp; ( for <font color="Olive" title="b3">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X2</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">A</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font>  st ( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> or <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> ) holds <br/>( ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b1">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">A</font> ) &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Olive" title="b1">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">A</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">f2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X2</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> or <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> ) implies ( ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> ) &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> or <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f2</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> ) &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font>;<br/>
<a NAME="E2:19"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></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 <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font></span><br/><span class="kw">thus </span><a NAME="E1:19_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> implies <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:19_1_1_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">and </span><br/><a NAME="E4:19_1_1_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/><a NAME="E5:19_1_1_1"/>
<font color="Maroon" title="c10">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/><a NAME="E6:19_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:19_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="tietze.html#T14" target="_self" title="TIETZE:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E7:19_1_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:19_1_1_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:19_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X1</font><span class="kw"> such that </span><br/><a NAME="E4:19_1_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<a NAME="E5:19_1_1"/>
<font color="Maroon" title="c10">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/>
<a NAME="E6:19_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="tietze.html#T14" target="_self" title="TIETZE:th.14">Th14</a></span>;<br/>
<a NAME="E7:19_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:19_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font></span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> implies <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:19_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:19_2"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">and </span><br/><a NAME="E4:19_2"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/><a NAME="E5:19_2"/>
<font color="Maroon" title="c10">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/><a NAME="E6:19_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:19_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="tietze.html#T14" target="_self" title="TIETZE:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E7:19_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:19"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f2</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">X2</font><span class="kw"> such that </span><br/><a NAME="E7:19"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<a NAME="E8:19"/>
<font color="Maroon" title="c10">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T55" title="PRE_TOPC:th.55">PRE_TOPC:55</a></span>;<br/>
<a NAME="E9:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="tietze.html#T14" target="_self" title="TIETZE:th.14">Th14</a></span>;<br/>
<a NAME="E10:19"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:19"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="tmap_1.html#K10" title="TMAP_1:func.10">union</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E9:19"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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