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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>



<b>assume </b><a NAME="E1:138"/>
<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:()"><i><font color="Red">::  thesis: </font></i></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="E2:138"/><b>then </b><i><font color="Green" title="E62">A1</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="partfun1.html#T128" title="PARTFUN1:th.128">PARTFUN1:128</a></i>;<br/>
<a NAME="E3:138"/><b>then </b><i><font color="Green" title="E63">A2</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E4:138"/>
( <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> &amp; <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> )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></i>;<br/>
<a NAME="E5:138"/><b>then </b><i><font color="Green" title="E64">A3</font></i>: 
(  <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> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E1:138_1"/><i><font color="Green" title="E65">A4</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E2:138_1"/><b>then </b><i><font color="Green" title="E66">A5</font></i>: 
( <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> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><b>then reconsider </b><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> <b>by </b><i><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="monoid_0.html#E2:138"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="monoid_0.html#E2:138_1"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></i>;<br/><b>thus </b><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>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:138_1"><i><font color="Green" title="E65">A4</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c4">x1</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c5">x2</font>
<b>by </b><i><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></i>
<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>
<b>by </b><i><a class="ref" href="monoid_0.html#D37" target="_self" title="MONOID_0:def.37">Def37</a></i>
<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>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:138_1"><i><font color="Green" title="E65">A4</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:138"/>
<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>)
 <b>by </b><i><a class="txt" href="monoid_0.html#E3:138"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="monoid_0.html#E5:138"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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