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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being 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> <br/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b3">h</font>, <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"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Olive" title="b1">M</font>  st <font color="Olive" title="b3">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b3">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Olive" title="b2">f</font> <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> &amp; <font color="Olive" title="b4">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b4">g</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Olive" title="b2">f</font> <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> holds <br/><font color="Olive" title="b3">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">h</font>, <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"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b2">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b2">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Olive" title="b1">f</font> <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> &amp; <font color="Olive" title="b3">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b3">g</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Olive" title="b1">f</font> <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> holds <br/><font color="Olive" title="b2">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h</font>, <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"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b1">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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> &amp; <font color="Olive" title="b2">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b2">g</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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> holds <br/><font color="Olive" title="b1">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">h</font>, <font color="Maroon" title="c5">g</font> be   <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>,<font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Maroon" title="c4">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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> &amp; <font color="Maroon" title="c5">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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> implies <font color="Maroon" title="c4">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:88"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c4">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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> or  not <font color="Maroon" title="c5">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  or  not <font color="Maroon" title="c5">g</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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> or <font color="Maroon" title="c4">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:88"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  or  not <font color="Maroon" title="c5">g</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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> or <font color="Maroon" title="c4">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:88"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c5">g</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">g</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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> or <font color="Maroon" title="c4">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:88"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">w</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>  st  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font>;<br/>
<a NAME="E5:88"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_1"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:88_1"/>
for <font color="Olive" title="b1">w</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>  st  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">w</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> implies <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_1_1"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font></span><br/>

<a NAME="E2:88_1_1"/><span class="lab"><font color="Green" title="E55">A8</font></span>: 
( <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c7">w</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T32" target="_self" title="ALGSTR_4:th.32">Th32</a></span>;<br/>
<a NAME="E3:88_1_1"/><span class="kw">then </span>
(  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E4:88_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A9</font></span>: 
(  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:88_1_1_1"/>
(  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_1_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:88_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">w</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/><a NAME="E2:88_1_1_1_1"/>
<font color="Maroon" title="c7">w</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">w9</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">w9</font> is   <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 href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Olive" title="b1">w9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1_1_1_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a></span>;<br/><a NAME="E3:88_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A11</font></span>: 
<font color="Maroon" title="c7">w</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> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T30" target="_self" title="ALGSTR_4:th.30">Th30</a></span>;<br/><a NAME="E4:88_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A12</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <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="c4">h</font> &amp; <font color="Maroon" title="c4">h</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c3">f</font> <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#D2" target="_self" title="ALGSTR_4:def.2">Def3</a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>;<br/><a NAME="E5:88_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A13</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <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="c5">g</font> &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c3">f</font> <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#D2" target="_self" title="ALGSTR_4:def.2">Def3</a>, <a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E51">A4</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E61">A14</font></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="c1">X</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="c7">w</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</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="c7">w</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </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="#E3:88_1_1_1_1"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E42">Lm2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" title="ALGSTR_4:def.18">Def19</a>, <a class="txt" href="#E2:88_1_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E1:88_1_1_1_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a></span>
;<br/><a NAME="E7:88_1_1_1_1"/>
<font color="Maroon" title="c7">w</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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88_1_1_1_1"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E42">Lm2</font></span></a></span>;<br/><a NAME="E8:88_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN: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#K16" title="ALGSTR_4:func.16">canon_image</a> <font color="Maroon" title="c1">X</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88_1_1_1_1"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E9:88_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A15</font></span>: 
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E10:88_1_1_1_1"/>
<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="c3">f</font>
 <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="E11:88_1_1_1_1"/><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> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E42">Lm2</font></span></a></span>;<br/><a NAME="E12:88_1_1_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">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> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E13:88_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:88_1_1_1_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/><a NAME="E14:88_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span> &amp; <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_1_1_1_1"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="txt" href="#E5:88_1_1_1_1"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E15:88_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <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">w</font> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <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">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_1_1_1_1"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="txt" href="#E5:88_1_1_1_1"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E16:88_1_1_1_1"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:88_1_1_1_2"/>
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">w1</font>, <font color="Maroon" title="c9">w2</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><span class="kw"> such that </span><br/><a NAME="E3:88_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
( <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">w2</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c8">w1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c9">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c7">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T34" target="_self" title="ALGSTR_4:th.34">Th34</a></span>;<br/><a NAME="E4:88_1_1_1_2"/>
( <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w1</font> &amp; <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E1:88_1_1"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="#E3:88_1_1_1_2"><span class="lab"><font color="Green" title="E57">A16</font></span></a></span>;<br/><a NAME="E5:88_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">w2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w1</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="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">w2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:88_1_1_1_2"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88_1_1_1_2"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E6:88"/><span class="lab"><font color="Green" title="E53">A17</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E52">A5</font></span></a>);<br/></span>
<a NAME="E7:88"/><span class="lab"><font color="Green" title="E54">A18</font></span>: 
for <font color="Olive" title="b1">w</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="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">w</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">k</font> =  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c6">w</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E2:88_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E53">A17</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:88_2"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:88"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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 <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 <br/><font color="Maroon" title="c4">h</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="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88"><span class="lab"><font color="Green" title="E54">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:88"/>
<font color="Maroon" title="c4">h</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
