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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><br/>  for <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font>  st <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">g</font> holds <br/> <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Olive" title="b2">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</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">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font>  st <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">g</font> holds <br/> <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Olive" title="b1">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</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="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">g</font> implies  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c5">g</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:95"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</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="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">g</font> or  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:95"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">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">  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c5">g</font></span><br/>

<a NAME="E3:95"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c3">Z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="algstr_4.html#T27" target="_self" onmouseover="rs('algstr_4/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f9</font> =  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c3">Z</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" onmouseover="rs('funct_2/T7')" onmouseout="rh()">FUNCT_2:7</a></span>;<br/>
<a NAME="E5:95"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  <font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="95_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" 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="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">v</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">w</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">v9</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c12">w9</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c3">Z</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95"><span class="lab"><font color="Green" title="E53">A3</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E54">A4</font></span>:  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c11">v9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>
;<br/>
<span class="lab"><font color="Green" title="E55">A5</font></span>:  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">w9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>
<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>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">v9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">w9</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">v9</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c11">v9</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">w9</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T31" target="_self" onmouseover="rs('algstr_4/T31')" onmouseout="rh()">Th31</a>, <a class="txt" href="#E2:95_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E3:95_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T31" target="_self" onmouseover="rs('algstr_4/T31')" onmouseout="rh()">Th31</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="E6:95"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c6">f9</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>;<br/>
<a NAME="E7:95"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</font>
 
;<br/>
<a NAME="E8:95"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">g</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#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E41">Lm3</font></span></a></span>;<br/>
<a NAME="E9:95"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A7</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#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E10:95"/>
<font color="Maroon" title="c1">X</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="c5">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="E11:95"/><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#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E41">Lm3</font></span></a>, <a class="txt" href="#E9:95"><span class="lab"><font color="Green" title="E55">A7</font></span></a></span>;<br/>
<a NAME="E12:95"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E13:95"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A8</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E14:95"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
;<br/>
<a NAME="E15:95"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:95"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E16:95"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A9</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></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="c6">f9</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="E17:95"/>
 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="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c6">f9</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="95_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> implies <font color="Maroon" title="c6">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:95_2"/><span class="lab"><font color="Green" title="E58">A10</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></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="c6">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:95_2"/>
 <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D21" target="_self" onmouseover="rs('algstr_4/D21')" onmouseout="rh()">Def21</a></span>;<br/>
<a NAME="E3:95_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A11</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></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> <span class="p1">(<span class="default"><a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font> )
 
;<br/>
<a NAME="E4:95_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
;<br/>
<a NAME="E5:95_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A12</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">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#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E41">Lm3</font></span></a></span>;<br/>
<a NAME="E6:95_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/>
<a NAME="E7:95_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A13</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">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#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E41">Lm3</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E62">A14</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#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:95_2"><span class="lab"><font color="Green" title="E60">A12</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 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#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:95_2"><span class="lab"><font color="Green" title="E61">A13</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="E9:95_2"/>
 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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="95_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:95_2_2"/><span class="lab"><font color="Green" title="E63">A15</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span><br/>

<a NAME="E2:95_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A16</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E3:95_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A17</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E41">Lm3</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_2_2"><span class="lab"><font color="Green" title="E63">A15</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<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="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95_2_2"><span class="lab"><font color="Green" title="E64">A16</font></span></a>, <a class="ref" href="algstr_4.html#D19" target="_self" onmouseover="rs('algstr_4/D19')" onmouseout="rh()">Def19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E3:95_2_2"><span class="lab"><font color="Green" title="E65">A17</font></span></a>, <a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E41">Lm3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:95_2"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="txt" href="#E1:95_2_2"><span class="lab"><font color="Green" title="E63">A15</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</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="E10:95_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c2">Y</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E8:95_2"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:95_2"/>
<font color="Maroon" title="c6">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_2"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="txt" href="#E3:95_2"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="partfun1.html#T53" onmouseover="rs('partfun1/T53')" onmouseout="rh()">PARTFUN1:53</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="E18:95"/><span class="kw">then </span>
<font color="Maroon" title="c6">f9</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:95"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="partfun1.html#T53" onmouseover="rs('partfun1/T53')" onmouseout="rh()">PARTFUN1:53</a></span>;<br/>
<a NAME="E19:95"/><span class="kw">then </span>
<font color="Maroon" title="c6">f9</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c3">Z</font>,1)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:95"><span class="lab"><font color="Green" title="E57">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:95"/>
 <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="algstr_4.html#K18" title="ALGSTR_4:func.18">free_magmaF</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="algstr_4.html#D21" target="_self" onmouseover="rs('algstr_4/D21')" onmouseout="rh()">Def21</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>
