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<div class="add">

<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">B</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="b2">N</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b2">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b2">N</font></span>)</span>,<font color="Olive" title="b2">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>,<font color="Olive" title="b1">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Olive" title="b2">N</font></span>)</span> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</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">N</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b1">N</font></span>)</span>,<font color="Olive" title="b1">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>,<font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>,<font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">f</font> =  <a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">g</font> = <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">I</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">J</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> ;<br/>
<a NAME="E1:110"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</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="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/>
<a NAME="E2:110"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</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="c1">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E3:110"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</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="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E4:110"/><span class="lab"><font color="Green" title="E64">A4</font></span>: 
<font color="Maroon" title="c3">N</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T78" title="GROUP_4:th.78">GROUP_4:78</a></span>;<br/>
<a NAME="E5:110"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A5</font></span>: 
<font color="Maroon" title="c3">N</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D1" target="_self" title="GROUP_6:def.1">Def1</a></span>;<br/>
<a NAME="E6:110"/><span class="lab"><font color="Green" title="E66">A6</font></span>: 
<font color="Maroon" title="c2">B</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T78" title="GROUP_4:th.78">GROUP_4:78</a></span>;<br/>
<a NAME="E7:110"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span>)</span> <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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span>)</span> or <font color="Maroon" title="c8">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:110_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span>)</span>
 ; <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="c8">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:110_1"/><span class="lab"><font color="Green" title="E67">A7</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:110_1"/><span class="lab"><font color="Green" title="E68">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E3:110_1"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">x9</font> = <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x99</font> = <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></span>;<br/>
<a NAME="E8:110_1"/><span class="lab"><font color="Green" title="E69">A9</font></span>: 
( <font color="Maroon" title="c12">x99</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c3">N</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c12">x99</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="group_6.html#T2" target="_self" title="GROUP_6:th.2">Th2</a></span>;<br/>
<span class="lab"><font color="Green" title="E70">A10</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<font color="Maroon" title="c11">x9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D9" target="_self" title="GROUP_6:def.9">Def9</a></span>
;<br/>
<a NAME="E10:110_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</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="c3">N</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T140" title="GROUP_3:th.140">GROUP_3:140</a></span>;<br/>
<a NAME="E11:110_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="txt" href="#E9:110_1"><span class="lab"><font color="Green" title="E70">A10</font></span></a>, <a class="txt" href="#E8:110_1"><span class="lab"><font color="Green" title="E69">A9</font></span></a>, <a class="ref" href="group_6.html#T28" target="_self" title="GROUP_6:th.28">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:110_1"/>
<font color="Maroon" title="c8">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></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>; <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<a NAME="E9:110"/><span class="lab"><font color="Green" title="E67">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="ref" href="group_6.html#T34" target="_self" title="GROUP_6:th.34">Th34</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</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="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">a9</font> = <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c12">b9</font> = <font color="Maroon" title="c10">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></span>;<br/><a NAME="E2:110_2"/><span class="lab"><font color="Green" title="E68">A12</font></span>: 
( <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><a NAME="E3:110_2"/>
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a9</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:110"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="txt" href="#E2:110_2"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></span>
;<br/> <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></span>;<br/>
<a NAME="E12:110"/><span class="lab"><font color="Green" title="E68">A13</font></span>: 
 <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c9">g</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_2.html#D6" title="GROUP_2:def.6">GROUP_2:def 6</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="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c9">g</font> or <font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font> ) &amp; (  not <font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font> or <font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c9">g</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c10">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></span>;<br/>
<span class="lab"><font color="Green" title="E69">A14</font></span>: <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c11">c</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D9" target="_self" title="GROUP_6:def.9">Def9</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E3:110_3"/>
( <font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c9">g</font> implies <font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font> )
  <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="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font> or <font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c9">g</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_3_2"/>
<font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c9">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="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font></span><br/>

<span class="kw">then </span><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T50" target="_self" title="GROUP_6:th.50">Th50</a></span>
<br/>.= 
 <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T29" target="_self" title="GROUP_6:th.29">Th29</a></span>
<br/>.= 
 <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="ref" href="group_6.html#D1" target="_self" title="GROUP_6:def.1">Def1</a></span>
;<br/>
<a NAME="E3:110_3_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_3"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="group_2.html#T136" title="GROUP_2:th.136">GROUP_2:136</a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:110_3_2"/>
<font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T99" title="GROUP_2:th.99">GROUP_2:99</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="E4:110_3"/>
<font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</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="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c9">g</font></span><br/>

<a NAME="E5:110_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T99" title="GROUP_2:th.99">GROUP_2:99</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c11">c</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font> = 
 <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="group_2.html#T136" title="GROUP_2:th.136">GROUP_2:136</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T29" target="_self" title="GROUP_6:th.29">Th29</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:110_3"/>
<font color="Maroon" title="c10">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c9">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_3"><span class="lab"><font color="Green" title="E69">A14</font></span></a>, <a class="ref" href="group_6.html#T50" target="_self" title="GROUP_6:th.50">Th50</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>
<a NAME="E13:110"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</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="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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:110_4"/>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span>
 ; <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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">g</font></span><br/>

<a NAME="E2:110_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b</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="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:110_4"/><span class="lab"><font color="Green" title="E69">A15</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:110_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T28" target="_self" title="GROUP_6:th.28">Th28</a></span>;<br/>
<a NAME="E6:110_4"/>
( <font color="Maroon" title="c2">B</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c11">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_5.html#T8" title="GROUP_5:th.8">GROUP_5:8</a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a1</font>, <font color="Maroon" title="c13">a2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E8:110_4"/><span class="lab"><font color="Green" title="E70">A16</font></span>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a2</font>
 <span class="kw">and </span><br/><a NAME="E9:110_4"/><span class="lab"><font color="Green" title="E71">A17</font></span>: 
<font color="Maroon" title="c12">a1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E10:110_4"/><span class="lab"><font color="Green" title="E72">A18</font></span>: 
<font color="Maroon" title="c13">a2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_5.html#T5" title="GROUP_5:th.5">GROUP_5:5</a></span>;<br/>
<a NAME="E11:110_4"/><span class="lab"><font color="Green" title="E73">A19</font></span>: 
<font color="Maroon" title="c12">a1</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 <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:110_4"><span class="lab"><font color="Green" title="E71">A17</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a2</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E4:110_4"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="txt" href="#E8:110_4"><span class="lab"><font color="Green" title="E70">A16</font></span></a>, <a class="ref" href="group_6.html#T2" target="_self" title="GROUP_6:th.2">Th2</a></span>
<br/>.= 
<font color="Maroon" title="c12">a1</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a2</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T127" title="GROUP_2:th.127">GROUP_2:127</a></span>
<br/>.= 
<font color="Maroon" title="c12">a1</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:110_4"><span class="lab"><font color="Green" title="E72">A18</font></span></a>, <a class="ref" href="group_2.html#T136" title="GROUP_2:th.136">GROUP_2:136</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D9" target="_self" title="GROUP_6:def.9">Def9</a></span>
<br/>.= 
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:110_4"><span class="lab"><font color="Green" title="E73">A19</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:110_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="txt" href="#E11:110_4"><span class="lab"><font color="Green" title="E73">A19</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1: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>
<a NAME="E14:110"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E15:110"/><span class="kw">then </span>
<font color="Maroon" title="c9">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E16:110"/><span class="kw">then </span>
 <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T67" target="_self" title="GROUP_6:th.67">Th67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:110"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N</font></span>)</span>,<font color="Maroon" title="c3">N</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span>,<font color="Maroon" title="c2">B</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="group_6.html#K2" title="GROUP_6:func.2">/\</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:110"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="group_6.html#E107"><span class="lab"><font color="Green" title="E60">Lm3</font></span></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>
