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<span class="kw">let </span><font color="Maroon" title="c1">x</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"> for <font color="Olive" title="b1">G</font> being  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/> for <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font> holds <br/> ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b3">H2</font> iff ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">H2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">H1</font>, <font color="Olive" title="b2">H2</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> holds <br/> ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b2">H2</font> iff ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">H2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">H1</font>, <font color="Maroon" title="c4">H2</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</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="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">H2</font> iff ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font> ) )</span><br/>







<span class="kw">thus </span><a NAME="E1:104"/>
( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">H2</font> implies ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</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="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font> implies <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">H2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:104_1"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">H2</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="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font> )</span><br/>

<a NAME="E2:104_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">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">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E3:104_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D10" target="_self" title="GROUP_2:def.10">Def10</a></span>;<br/>
<a NAME="E4:104_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">H2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:104_1"/>
( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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">assume </span><a NAME="E2:104"/>
( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</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="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">H2</font></span><br/>

<a NAME="E3:104"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">H2</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E4:104"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:104"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D10" target="_self" title="GROUP_2:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:104"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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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