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<b>let </b><font color="Maroon" title="c1">GX</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">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">A</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> &amp; <font color="Olive" title="b2">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> holds <br/>( <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> &amp; <font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> implies ( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> ) )</span><br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">Fa</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E3:24"/><i><font color="Green" title="E19">A2</font></i>: 
( ( for <font color="Olive" title="b1">Ba</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">Ba</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> holds <br/><font color="Olive" title="b1">Ba</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font> ) &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">Fa</font> )
 <b>by </b><i><a class="ref" href="connsp_3.html#D2" target="_self" title="CONNSP_3:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">Fb</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E5:24"/><i><font color="Green" title="E20">A3</font></i>: 
( ( for <font color="Olive" title="b1">Bb</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">Bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Fb</font> holds <br/><font color="Olive" title="b1">Bb</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font> ) &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">Fb</font> )
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:24"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="connsp_3.html#D2" target="_self" title="CONNSP_3:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Fc</font> = <font color="Maroon" title="c4">Fa</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Fb</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font> ;<br/>
<a NAME="E7:24"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">Fc</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E3:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="connsp_3.html#E5:24"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></i>;<br/>
<a NAME="E8:24"/><i><font color="Green" title="E22">A5</font></i>: 
for <font color="Olive" title="b1">B2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Fb</font> holds <br/><font color="Olive" title="b1">B2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">B2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Fb</font> implies <font color="Maroon" title="c7">B2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font> )</span><br/>

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon" title="c7">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Fb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">B2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font></span><br/>

<a NAME="E2:24_1"/><b>then </b>
( <font color="Maroon" title="c7">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> or <font color="Maroon" title="c7">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Fb</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1"/>
<font color="Maroon" title="c7">B2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E3:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="connsp_3.html#E5:24"><i><font color="Green" title="E20">A3</font></i></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="E9:24"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c7">Fd</font> = <font color="Maroon" title="c4">Fa</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Fb</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font> ;<br/>
<a NAME="E2:24_2"/><i><font color="Green" title="E23">A6</font></i>: 
for <font color="Olive" title="b1">B4</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">B4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Fd</font> holds <br/><font color="Olive" title="b1">B4</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">B4</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">B4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Fd</font> implies <font color="Maroon" title="c8">B4</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font> )</span><br/>

<b>assume </b><a NAME="E1:24_2_1"/>
<font color="Maroon" title="c8">B4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Fd</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">B4</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font></span><br/>

<a NAME="E2:24_2_1"/><b>then </b>
( <font color="Maroon" title="c8">B4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> &amp; <font color="Maroon" title="c8">B4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Fb</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:24_2_1"/>
<font color="Maroon" title="c8">B4</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E3:24"><i><font color="Green" title="E19">A2</font></i></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="E3:24_2"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">Fd</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:24_2_2"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</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> <font color="Maroon" title="c7">Fd</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c8">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="c7">Fd</font> )</span><br/>

<b>assume </b><a NAME="E1:24_2_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c7">Fd</font></span><br/>

<a NAME="E2:24_2_2_1"/><b>then </b><i><font color="Green" title="E25">A8</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">F1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:24_2_2_1"/><i><font color="Green" title="E26">A9</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F1</font> &amp; <font color="Maroon" title="c9">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> )
 <b>by </b><i><a class="txt" href="connsp_3.html#E3:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">F2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:24_2_2_1"/><i><font color="Green" title="E27">A10</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> &amp; <font color="Maroon" title="c10">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Fb</font> )
 <b>by </b><i><a class="txt" href="connsp_3.html#E5:24"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="connsp_3.html#E2:24_2_2_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:24_2_2_1"/><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c9">F1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">F2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="connsp_3.html#E4:24_2_2_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="connsp_3.html#E6:24_2_2_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">C1</font> = <font color="Maroon" title="c9">F1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> <b>by </b><i><a class="txt" href="connsp_3.html#E4:24_2_2_1"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">C2</font> = <font color="Maroon" title="c10">F2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> <b>by </b><i><a class="txt" href="connsp_3.html#E6:24_2_2_1"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<a NAME="E10:24_2_2_1"/><i><font color="Green" title="E29">A12</font></i>: 
<font color="Maroon" title="c11">C1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E3:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="connsp_3.html#E4:24_2_2_1"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>
<a NAME="E11:24_2_2_1"/>
<font color="Maroon" title="c12">C2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E5:24"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="connsp_3.html#E6:24_2_2_1"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<a NAME="E12:24_2_2_1"/><b>then </b>
( <font color="Maroon" title="c11">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">C2</font> or <font color="Maroon" title="c11">C1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">C2</font> )
 
<b>by </b><i><a class="txt" href="connsp_3.html#E10:24_2_2_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="connsp_1.html#T37" title="CONNSP_1:th.37">CONNSP_1:37</a></i>;<br/>
<a NAME="E13:24_2_2_1"/><b>then </b>
<font color="Maroon" title="c11">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Fb</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E4:24_2_2_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="connsp_3.html#E6:24_2_2_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="txt" href="connsp_3.html#E7:24_2_2_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E14:24_2_2_1"/>
<font color="Maroon" title="c8">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="c7">Fd</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E4:24_2_2_1"><i><font color="Green" title="E26">A9</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/>


</div><b>end;</b></div>
<a NAME="E2:24_2_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">Fd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="c8">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="c7">Fd</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<b>assume </b><a NAME="E1:24_2_2_2"/>
<font color="Maroon" title="c8">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="c7">Fd</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">X4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:24_2_2_2"/><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X4</font> &amp; <font color="Maroon" title="c9">X4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Fd</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:24_2_2_2"/><i><font color="Green" title="E26">A14</font></i>: 
( <font color="Maroon" title="c9">X4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Fa</font> &amp; <font color="Maroon" title="c9">X4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Fb</font> )
 
<b>by </b><i><a class="txt" href="connsp_3.html#E3:24_2_2_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:24_2_2_2"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c8">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">Fa</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E3:24_2_2_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:24_2_2_2"/>
<font color="Maroon" title="c8">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="c5">Fb</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E3:24_2_2_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="connsp_3.html#E4:24_2_2_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:24_2_2_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E3:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="connsp_3.html#E5:24"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="connsp_3.html#E5:24_2_2_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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>
<b>hence </b><a NAME="E3:24_2_2"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">Fd</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:24_2_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
<b>hence </b><a NAME="E4:24_2"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E2:24_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="connsp_3.html#D2" target="_self" title="CONNSP_3:def.2">Def2</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>
<b>hence </b><a NAME="E10:24"/>
( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is    <a href="connsp_3.html#M1" title="CONNSP_3:mode.1">a_union_of_components</a> of <font color="Maroon" title="c1">GX</font> )
 <b>by </b><i><a class="txt" href="connsp_3.html#E7:24"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="connsp_3.html#E8:24"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="connsp_3.html#D2" target="_self" title="CONNSP_3:def.2">Def2</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>
