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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">R</font> being  <a href="algstr_4.html#V1" title="ALGSTR_4:attr.1">compatible</a>  <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Olive" title="b1">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> <font color="Maroon" title="c1">X</font>,1</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="algstr_4.html#V1" title="ALGSTR_4:attr.1">compatible</a>  <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>; <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"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> <font color="Maroon" title="c1">X</font>,1</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">A</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> <font color="Maroon" title="c1">X</font>,1</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> <font color="Maroon" title="c1">X</font>,1</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E2:84"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c4">A</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 <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T36" target="_self" title="ALGSTR_4:th.36">Th37</a></span>;<br/>
<a NAME="E3:84"/>
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"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font></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> <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span>
 
<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="E4:84"/><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"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/>
<a NAME="E5:84"/><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"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:84"/><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"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</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 <span class="p1">(<span class="default"><a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T15" target="_self" title="ALGSTR_4:th.15">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:84"/>
<span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> <font color="Maroon" title="c1">X</font>,1</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T7" target="_self" title="ALGSTR_4:th.7">Th7</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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