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<span class="kw">let </span><font color="Maroon" title="c1">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="c1">G</font><br/> for <font color="Olive" title="b3">H3</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b3">H3</font> &amp; <font color="Olive" title="b2">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b3">H3</font> holds <br/><font color="Olive" title="b1">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b2">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b3">H3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</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">H3</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c2">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">H3</font> &amp; <font color="Maroon" title="c3">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">H3</font> holds <br/><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">H3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">H3</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">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="c2">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">H3</font> &amp; <font color="Maroon" title="c3">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">H3</font> implies <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">H3</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:91"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
( <font color="Maroon" title="c2">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">H3</font> &amp; <font color="Maroon" title="c3">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">H3</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="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">H3</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">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="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font> implies <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H3</font> )</span><br/><span class="kw">assume </span><a NAME="E1:91_1"/>
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">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="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H3</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c7">I</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <span class="kw"> such that </span><br/><a NAME="E3:91_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">I</font>
 <span class="kw">and </span><br/><a NAME="E4:91_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:91_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T37" target="_self" title="GROUP_4:th.37">Th37</a></span>;<br/><a NAME="E6:91_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H3</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/><a NAME="E7:91_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">H3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E8:91_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">H3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E9:91_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</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">H3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E5:91_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="group_4.html#T37" target="_self" title="GROUP_4:th.37">Th37</a></span>;<br/><span class="kw">hence </span><a NAME="E10:91_1"/>
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T40" target="_self" title="GROUP_4:th.40">Th40</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">hence </span><a NAME="E3:91"/>
<font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">H3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T67" title="GROUP_2:th.67">GROUP_2:67</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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