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<span class="kw">let </span><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">N</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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 <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Olive" title="b1">R</font></span>)</span>,<font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b2">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<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#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<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> holds <br/><font color="Olive" title="b2">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">g2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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 <font color="Maroon" title="c1">M</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.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="c3">R</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.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="c3">R</font></span>)</span> holds <br/><font color="Olive" title="b1">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">g2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">N</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.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="c3">R</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.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="c3">R</font></span>)</span> implies <font color="Maroon" title="c4">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g2</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:90"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.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="c3">R</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.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="c3">R</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">Y</font> =  <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="c3">R</font></span>)</span>;<br/>
<a NAME="E2:90"/>
 <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="c3">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"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c3">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E3:90"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g1</font> <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="c3">R</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g2</font> <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="c3">R</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:90"/>
<font color="Maroon" title="c4">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="funct_1.html#T86" onmouseover="rs('funct_1/T86')" onmouseout="rh()">FUNCT_1:86</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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