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<span class="kw">let </span><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> implies  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font> is    <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font> )</span><br/>



<a NAME="E1:139"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>;<br/>
<a NAME="E2:139"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span></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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:139"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>;<br/>
<a NAME="E4:139"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font>)</span>)</span></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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:139"/>
<font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font> is    <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E6:139"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A3</font></span>: 
 <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T50" title="PARTFUN1:th.50">PARTFUN1:50</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E67">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <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"><span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:139_1"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <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"><span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><a NAME="E2:139_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A6</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>) &amp; <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c5">x2</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c7">y2</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:139"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E2:139_1"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>;<br/><span class="kw">thus </span><font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<font color="Maroon" title="c4">x1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:139_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="mcart_1.html#T21" title="MCART_1:th.21">MCART_1:21</a></span>
<br/>.= 
<font color="Maroon" title="c5">x2</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c4">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>
<br/>.= 
<font color="Maroon" title="c6">y1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">y2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font>) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:139_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="mcart_1.html#T21" title="MCART_1:th.21">MCART_1:21</a></span>
; <a class="txt" 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="E8:139"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">Y</font>,<font color="Maroon" title="c1">Y</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:139"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:139"/>
<font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c1">Y</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K12" title="MONOID_0:func.12">GPFuncs</a> <font color="Maroon" title="c2">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:139"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E4:139"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E7:139"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">MONOID_0:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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