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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">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font><br/>  for <font color="Olive" title="b2">H</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c2">N</font><br/>  for <font color="Olive" title="b3">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>  st <font color="Olive" title="b1">f</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H</font> <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="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Olive" title="b1">f</font> holds <br/> ex <font color="Olive" title="b4">g</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="b3">R</font></span>)</span>,<font color="Olive" title="b2">H</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b4">g</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="b3">R</font></span>)</span> &amp; <font color="Olive" title="b4">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b4">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">M</font>,<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">  for <font color="Olive" title="b1">H</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c2">N</font><br/>  for <font color="Olive" title="b2">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>  st <font color="Maroon" title="c3">f</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H</font> <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="c3">f</font> &amp; <font color="Olive" title="b2">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Maroon" title="c3">f</font> holds <br/> ex <font color="Olive" title="b3">g</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="b2">R</font></span>)</span>,<font color="Olive" title="b1">H</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b3">g</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="b2">R</font></span>)</span> &amp; <font color="Olive" title="b3">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b3">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">H</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <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">  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>  st <font color="Maroon" title="c3">f</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font> <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="c3">f</font> &amp; <font color="Olive" title="b1">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Maroon" title="c3">f</font> holds <br/> ex <font color="Olive" title="b2">g</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="c4">H</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">g</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> &amp; <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c3">f</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font> <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="c3">f</font> &amp; <font color="Maroon" title="c5">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Maroon" title="c3">f</font> implies  ex <font color="Olive" title="b1">g</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="c5">R</font></span>)</span>,<font color="Maroon" title="c4">H</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">g</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="c5">R</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:89"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</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"> (  not  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font> <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="c3">f</font> or  not <font color="Maroon" title="c5">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Maroon" title="c3">f</font> or  ex <font color="Olive" title="b1">g</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="c5">R</font></span>)</span>,<font color="Maroon" title="c4">H</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">g</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="c5">R</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:89"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font> <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="c3">f</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"> (  not <font color="Maroon" title="c5">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Maroon" title="c3">f</font> or  ex <font color="Olive" title="b1">g</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="c5">R</font></span>)</span>,<font color="Maroon" title="c4">H</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">g</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="c5">R</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:89"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c5">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Maroon" title="c3">f</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">  ex <font color="Olive" title="b1">g</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="c5">R</font></span>)</span>,<font color="Maroon" title="c4">H</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">g</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="c5">R</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">g</font> = <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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E4:89"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:89_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</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"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</font> or <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:89_1"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">z1</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" onmouseover="rs('relat_1/D8')" onmouseout="rh()">RELAT_1:def 8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:89_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</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="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:89_1"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c11">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" onmouseover="rs('relat_1/D8')" onmouseout="rh()">RELAT_1:def 8</a></span>;<br/>
<a NAME="E7:89_1"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">z1</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z2</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E6:89_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</a></span>;<br/>
<a NAME="E8:89_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">z1</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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <font color="Maroon" title="c11">z2</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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">z1</font> = <font color="Maroon" title="c10">z1</font>, <font color="Maroon" title="c13">z2</font> = <font color="Maroon" title="c11">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E10:89_1"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
( <font color="Maroon" title="c7">x</font> <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="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">z1</font> &amp; <font color="Maroon" title="c7">x</font> <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="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">z2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:89_1"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E11:89_1"/><span class="lab"><font color="Green" title="E55">A8</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E6:89_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E12:89_1"/>
( <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c12">z1</font>) &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c13">z2</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D6" target="_self" onmouseover="rs('algstr_4/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E13:89_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">z2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c12">z1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:89_1"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</a></span>;<br/>
<a NAME="E14:89_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c13">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:89_1"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:89_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="algstr_4.html#D8" target="_self" onmouseover="rs('algstr_4/D8')" onmouseout="rh()">Def8</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">g</font> = <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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">f</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D1" onmouseover="rs('funct_1/D1')" onmouseout="rh()">FUNCT_1:def 1</a></span>;<br/>
<a NAME="E6:89"/>
 <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="c5">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="c5">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="E7:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A9</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="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></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="c5">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T20" onmouseover="rs('relat_1/T20')" onmouseout="rh()">RELAT_1:20</a></span>;<br/>
<a NAME="E8:89"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<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/>
<a NAME="E9:89"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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="c5">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
;<br/>
<a NAME="E10:89"/><span class="kw">then </span>
 <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="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T20" onmouseover="rs('relat_1/T20')" onmouseout="rh()">RELAT_1:20</a></span>;<br/>
<a NAME="E11:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A10</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">g</font> <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="c5">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:89"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E12:89"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</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">M</font>
 
;<br/>
<a NAME="E13:89"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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="c5">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/>
<a NAME="E14:89"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T20" onmouseover="rs('relat_1/T20')" onmouseout="rh()">RELAT_1:20</a></span>;<br/>
<a NAME="E15:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">g</font> <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="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="relat_1.html#T28" onmouseover="rs('relat_1/T28')" onmouseout="rh()">RELAT_1:28</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">g</font> as   <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="c5">R</font></span>)</span>,<font color="Maroon" title="c4">H</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:89"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">g</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="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c8">g</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="c5">R</font></span>)</span> &amp; <font color="Maroon" title="c8">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Maroon" title="c8">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  )</span><br/>

<a NAME="E17:89"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</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> <font color="Maroon" title="c8">g</font> &amp; <font color="Olive" title="b2">x2</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> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c9">x1</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> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c10">x2</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> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font> implies <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:89_2"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<font color="Maroon" title="c9">x1</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> <font color="Maroon" title="c8">g</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"> (  not <font color="Maroon" title="c10">x2</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> <font color="Maroon" title="c8">g</font> or  not <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font> or <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:89_2"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<font color="Maroon" title="c10">x2</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> <font color="Maroon" title="c8">g</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"> (  not <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font> or <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:89_2"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</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="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>;<br/>
<a NAME="E4:89_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:89_2"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c12">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" onmouseover="rs('relat_1/D8')" onmouseout="rh()">RELAT_1:def 8</a></span>;<br/>
<a NAME="E7:89_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E2:89_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:89_2"/><span class="lab"><font color="Green" title="E58">A16</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c13">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">z2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" onmouseover="rs('relat_1/D8')" onmouseout="rh()">RELAT_1:def 8</a></span>;<br/>
<a NAME="E10:89_2"/><span class="lab"><font color="Green" title="E59">A17</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c9">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">z2</font>,<font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:89_2"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E9:89_2"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</a></span>;<br/>
<a NAME="E11:89_2"/><span class="kw">then </span>
( <font color="Maroon" title="c12">z1</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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <font color="Maroon" title="c13">z2</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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">z1</font> = <font color="Maroon" title="c12">z1</font>, <font color="Maroon" title="c15">z2</font> = <font color="Maroon" title="c13">z2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E13:89_2"/>
( <font color="Maroon" title="c14">z1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c15">z2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:89_2"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E9:89_2"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E14:89_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z1</font>,<font color="Maroon" title="c15">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K7" title="ALGSTR_4:func.7">equ_kernel</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D8" target="_self" onmouseover="rs('algstr_4/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E15:89_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A18</font></span>: 
<font color="Maroon" title="c15">z2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c14">z1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<a NAME="E16:89_2"/><span class="lab"><font color="Green" title="E61">A19</font></span>: 
( <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c14">z1</font>) &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c15">z2</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D6" target="_self" onmouseover="rs('algstr_4/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E17:89_2"/>
( <font color="Maroon" title="c9">x1</font> <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="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z1</font> &amp; <font color="Maroon" title="c10">x2</font> <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="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">z2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:89_2"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:89_2"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:89_2"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="txt" href="#E15:89_2"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</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/>


</div><span class="kw">end;</span></div>
<a NAME="E18:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A20</font></span>: 
<font color="Maroon" title="c8">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E19:89"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">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> <font color="Maroon" title="c3">f</font> iff ( <font color="Olive" title="b1">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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <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> <font color="Maroon" title="c3">f</font> iff ( <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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> ) )</span><br/>

<div><span class="kw">hereby </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="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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> implies <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> <font color="Maroon" title="c3">f</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:89_3_1"/>
<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> <font color="Maroon" title="c3">f</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="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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> )</span><br/><a NAME="E2:89_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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 <font color="Maroon" title="c1">M</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:89_3_1"/>
<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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">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>; <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"> <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font></span><br/><a NAME="E4:89_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c5">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E5:89_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c5">R</font></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E6:89_3_1"/>
<span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font>
 <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>; <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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:89_3"/>
( <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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</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="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> <font color="Maroon" title="c3">f</font></span><br/>

<a NAME="E3:89_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">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 <font color="Maroon" title="c1">M</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:89_3"/>
<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> <font color="Maroon" title="c3">f</font>
 <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>; <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/>


</div><span class="kw">end;</span></div>
<a NAME="E20:89"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <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> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c3">f</font> <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">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:89_4"/><span class="lab"><font color="Green" title="E56">A22</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> <font color="Maroon" title="c3">f</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="c3">f</font> <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">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">y</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>;<br/>
<a NAME="E2:89_4"/>
<span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_4"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="txt" href="#E19:89"><span class="lab"><font color="Green" title="E55">A21</font></span></a></span>;<br/>
<a NAME="E3:89_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:89_4"/><span class="lab"><font color="Green" title="E57">A23</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><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="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" onmouseover="rs('relat_1/D8')" onmouseout="rh()">RELAT_1:def 8</a></span>;<br/>
<a NAME="E6:89_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<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="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_4"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</a></span>;<br/>
<a NAME="E7:89_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A24</font></span>: 
( <font color="Maroon" title="c11">z</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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</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> <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E8:89_4"/><span class="lab"><font color="Green" title="E59">A25</font></span>: 
( <font color="Maroon" title="c11">z</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> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_4"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">z9</font> = <font color="Maroon" title="c11">z</font>, <font color="Maroon" title="c13">x9</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="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_4"><span class="lab"><font color="Green" title="E56">A22</font></span></a></span>;<br/>
<a NAME="E10:89_4"/>
( <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c12">z9</font>) &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c13">x9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D6" target="_self" onmouseover="rs('algstr_4/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E11:89_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">z9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c13">x9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:89_4"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</a></span>;<br/>
<a NAME="E12:89_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:89_4"/>
<font color="Maroon" title="c3">f</font> <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">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:89_4"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="algstr_4.html#D8" target="_self" onmouseover="rs('algstr_4/D8')" onmouseout="rh()">Def8</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E21:89"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c8">g</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="c5">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:89"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="ref" href="funct_1.html#T10" onmouseover="rs('funct_1/T10')" onmouseout="rh()">FUNCT_1:10</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"> ( <font color="Maroon" title="c8">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Maroon" title="c8">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  )</span><br/>

<a NAME="E22:89"/>
<font color="Maroon" title="c8">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="txt" href="#E15:89"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:89"/>
<font color="Maroon" title="c8">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:89"><span class="lab"><font color="Green" title="E54">A20</font></span></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"> <font color="Maroon" title="c8">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> </span><br/>

<a NAME="E24:89"/>
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</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="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c5">R</font></span>)</span> holds  <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="Olive" title="b1">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w</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="Olive" title="b1">v</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="Olive" title="b2">w</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c10">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c5">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="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">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</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">v</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">w</font></span>)</span></span><br/>

<a NAME="E1:89_5"/>
<font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c5">R</font></span>)</span>
 
;<br/>
<a NAME="E2:89_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font>
 
<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/>
<a NAME="E3:89_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:89_5"/><span class="lab"><font color="Green" title="E56">A26</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font></span>)</span>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" onmouseover="rs('relat_1/D8')" onmouseout="rh()">RELAT_1:def 8</a></span>;<br/>
<a NAME="E6:89_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_5"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</a></span>;<br/>
<a NAME="E7:89_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A27</font></span>: 
( <font color="Maroon" title="c11">z</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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E8:89_5"/><span class="lab"><font color="Green" title="E58">A28</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_5"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E9:89_5"/>
<font color="Maroon" title="c9">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c5">R</font></span>)</span>
 
;<br/>
<a NAME="E10:89_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font>
 
<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/>
<a NAME="E11:89_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">v</font>,<span class="p2">(<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">v</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E13:89_5"/><span class="lab"><font color="Green" title="E59">A29</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c12">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<span class="p2">(<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">v</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" onmouseover="rs('relat_1/D8')" onmouseout="rh()">RELAT_1:def 8</a></span>;<br/>
<a NAME="E14:89_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c9">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89_5"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</a></span>;<br/>
<a NAME="E15:89_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A30</font></span>: 
( <font color="Maroon" title="c12">z1</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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E16:89_5"/><span class="lab"><font color="Green" title="E61">A31</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z1</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">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89_5"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E17:89_5"/>
<font color="Maroon" title="c10">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c5">R</font></span>)</span>
 
;<br/>
<a NAME="E18:89_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font>
 
<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/>
<a NAME="E19:89_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">w</font>,<span class="p2">(<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">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E21:89_5"/><span class="lab"><font color="Green" title="E62">A32</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c13">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">R</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">z2</font>,<span class="p2">(<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">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" onmouseover="rs('relat_1/D8')" onmouseout="rh()">RELAT_1:def 8</a></span>;<br/>
<a NAME="E22:89_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">z2</font>,<font color="Maroon" title="c10">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:89_5"><span class="lab"><font color="Green" title="E62">A32</font></span></a>, <a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</a></span>;<br/>
<a NAME="E23:89_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A33</font></span>: 
( <font color="Maroon" title="c13">z2</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"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E24:89_5"/><span class="lab"><font color="Green" title="E64">A34</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z2</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">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:89_5"><span class="lab"><font color="Green" title="E62">A32</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">z1</font> = <font color="Maroon" title="c12">z1</font>, <font color="Maroon" title="c15">z2</font> = <font color="Maroon" title="c13">z2</font>, <font color="Maroon" title="c16">z</font> = <font color="Maroon" title="c11">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:89_5"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="txt" href="#E23:89_5"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="txt" href="#E7:89_5"><span class="lab"><font color="Green" title="E57">A27</font></span></a></span>;<br/>
<a NAME="E26:89_5"/><span class="lab"><font color="Green" title="E65">A35</font></span>: 
<span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">z</font> <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="c5">R</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="c14">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:89_5"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="txt" href="#E23:89_5"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="txt" href="#E7:89_5"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>;<br/>
<a NAME="E27:89_5"/>
( <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</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="c14">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z2</font></span>)</span>) &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c5">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c16">z</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D6" target="_self" onmouseover="rs('algstr_4/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E28:89_5"/><span class="kw">then </span>
<font color="Maroon" title="c14">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c16">z</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:89_5"><span class="lab"><font color="Green" title="E65">A35</font></span></a>, <a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</a></span>;<br/>
<a NAME="E29:89_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E66">A36</font></span>: <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">z</font> = 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c15">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="algstr_4.html#D8" target="_self" onmouseover="rs('algstr_4/D8')" onmouseout="rh()">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z1</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="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>
;<br/>
<a NAME="E31:89_5"/><span class="lab"><font color="Green" title="E67">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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">v</font></span>)</span>,<span class="p2">(<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">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">thus </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">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">w</font></span>)</span> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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">v</font></span>)</span>,<span class="p2">(<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">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:89_5"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E24:89_5"><span class="lab"><font color="Green" title="E64">A34</font></span></a>, <a class="txt" href="#E30:89_5"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="txt" href="#E8:89_5"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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">v</font></span>)</span>,<span class="p2">(<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">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:89_5"><span class="lab"><font color="Green" title="E67">A37</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.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">v</font></span>)</span>,<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">w</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">N</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.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">v</font></span>)</span>,<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">w</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="realset1.html#D2" onmouseover="rs('realset1/D2')" onmouseout="rh()">REALSET1:def 2</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">v</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">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D9" target="_self" onmouseover="rs('algstr_4/D9')" onmouseout="rh()">Def9</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E25:89"/>
<font color="Maroon" title="c8">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</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/>


</div>
