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<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <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"> for <font color="Olive" title="b1">IL</font> being 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> <br/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Olive" title="b1">IL</font>,<font color="Maroon" title="c1">N</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="ami_1.html#M6" title="AMI_1:mode.6">IL-Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Olive" title="b2">S</font>  st <font color="Olive" title="b3">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> holds <br/>( ( for <font color="Olive" title="b4">m</font>, <font color="Olive" title="b5">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">n</font> iff <font color="Olive" title="b3">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b4">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Olive" title="b3">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b5">n</font> ) ) iff for <font color="Olive" title="b4">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b3">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b4">k</font></span>)</span> &amp; ( for <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b4">k</font></span>)</span> holds <br/><font color="Olive" title="b4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">j</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">IL</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c2">IL</font>,<font color="Maroon" title="c1">N</font><br/> for <font color="Olive" title="b2">f</font> being   <a href="ami_1.html#M6" title="AMI_1:mode.6">IL-Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> holds <br/>( ( for <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">n</font> iff <font color="Olive" title="b2">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b3">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Olive" title="b2">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b4">n</font> ) ) iff for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b2">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b3">k</font></span>)</span> &amp; ( for <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b3">k</font></span>)</span> holds <br/><font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">j</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c2">IL</font>,<font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="ami_1.html#M6" title="AMI_1:mode.6">IL-Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c3">S</font>  st <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> holds <br/>( ( for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> iff <font color="Olive" title="b1">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Olive" title="b1">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b3">n</font> ) ) iff for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">k</font></span>)</span> &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">k</font></span>)</span> holds <br/><font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="ami_1.html#M6" title="AMI_1:mode.6">IL-Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> implies ( ( for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> iff <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">n</font> ) ) iff for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) ) ) )</span><br/>







<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> iff <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">n</font> ) ) iff for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) ) ) implies for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> iff <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">n</font> ) )</span>
<div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E13">A2</font></i>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> iff <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font></span>)</span> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font></span>)</span> holds <br/><font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> ) )</span><br/><a NAME="E2:27_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:27_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:27_1"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">F</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#M5" title="AMI_1:mode.5">IL-FinSequence</a> of <font color="Maroon" title="c3">S</font><b> such that </b><br/><a NAME="E5:27_1"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E6:27_1"/><i><font color="Green" title="E15">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> holds <br/><font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="amistd_1.html#D8" target="_self" title="AMISTD_1:def.8">Def8</a></i>;<br/><a NAME="E7:27_1"/><i><font color="Green" title="E16">A5</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E8:27_1"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:27"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E9:27_1"/><i><font color="Green" title="E18">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:27_1"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1"/>
not <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:27_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="amistd_1.html#E8:27_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="amistd_1.html#E9:27_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1"/>
contradiction
 ; <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><a NAME="E11:27_1"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E5:27_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="revrot_1.html#T3" title="REVROT_1:th.3">REVROT_1:3</a></i>;<br/><b>set </b><font color="Maroon" title="c7">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font>;<br/><a NAME="E12:27_1"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></i>;<br/><a NAME="E13:27_1"/><i><font color="Green" title="E22">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/><a NAME="E14:27_1"/><b>then </b><i><font color="Green" title="E23">A12</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:27_1"/><i><font color="Green" title="E24">A13</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E7:27_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_1"/><b>then </b>
<font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E17:27_1"/><b>then </b><i><font color="Green" title="E25">A14</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E5:27_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="amistd_1.html#E10:27_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="amistd_1.html#E12:27_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="amistd_1.html#E14:27_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c8">F1</font> = <font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E18:27_1"/><i><font color="Green" title="E26">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="finseq_4.html#T46" title="FINSEQ_4:th.46">FINSEQ_4:46</a></i>;<br/><a NAME="E19:27_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font>
 ;<br/><a NAME="E20:27_1"/><b>then </b><i><font color="Green" title="E27">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> )
 <b>by </b><i><a class="txt" href="amistd_1.html#E14:27_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="amistd_1.html#E17:27_1"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E28">A17</font></i>: <font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font></span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E13:27_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="amistd_1.html#E18:27_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></i>
;<br/><a NAME="E22:27_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="amistd_1.html#E5:27_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="amistd_1.html#E10:27_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="amistd_1.html#E12:27_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="amistd_1.html#E15:27_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="amistd_1.html#E18:27_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E23:27_1"/><b>then </b>
<font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">IL</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="finseq_4.html#T53" title="FINSEQ_4:th.53">FINSEQ_4:53</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">F1</font> = <font color="Maroon" title="c6">F</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">-|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#M5" title="AMI_1:mode.5">IL-FinSequence</a> of <font color="Maroon" title="c3">S</font> <b>by </b><i><a class="ref" href="ami_1.html#D34" title="AMI_1:def.34">AMI_1:def 34</a></i>;<br/><a NAME="E25:27_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">IL</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E8:27_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E27:27_1"/><i><font color="Green" title="E29">A18</font></i>: 
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">m</font> = <font color="Maroon" title="c10">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="amistd_1.html#E27:27_1"><i><font color="Green" title="E29">A18</font></i></a></i>;<br/><a NAME="E29:27_1"/><i><font color="Green" title="E30">A19</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F1</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E31">A20</font></i>: <font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> 1 = 
<font color="Maroon" title="c9">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="amistd_1.html#E29:27_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="finseq_4.html#T48" title="FINSEQ_4:th.48">FINSEQ_4:48</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="amistd_1.html#E5:27_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="amistd_1.html#E15:27_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><a NAME="E31:27_1"/><i><font color="Green" title="E32">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E32:27_1"/><i><font color="Green" title="E33">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F1</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E34">A23</font></i>: <font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span> = 
<font color="Maroon" title="c9">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="amistd_1.html#E32:27_1"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="finseq_4.html#T48" title="FINSEQ_4:th.48">FINSEQ_4:48</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E31:27_1"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font> implies <font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c12">n</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:27_1_5"/><i><font color="Green" title="E35">A24</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><br/><a NAME="E2:27_1_5"/><b>then </b><i><font color="Green" title="E36">A25</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_1_5"/><i><font color="Green" title="E37">A26</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font> )
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:27_1_5"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_1_5"/><b>then </b><i><font color="Green" title="E38">A27</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_1_5"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="amistd_1.html#E1:27_1_5"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:27_1_5"/><b>then </b><i><font color="Green" title="E39">A28</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:27_1_5"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:27_1_5"/>
<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="amistd_1.html#E3:27_1_5"><i><font color="Green" title="E37">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:27_1_5"/><b>then </b><i><font color="Green" title="E40">A29</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E3:27_1_5"><i><font color="Green" title="E37">A26</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><i><font color="Green" title="E41">A30</font></i>: <font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c9">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E4:27_1_5"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="amistd_1.html#E4:27_1_5"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="finseq_4.html#T48" title="FINSEQ_4:th.48">FINSEQ_4:48</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E8:27_1_5"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><i><font color="Green" title="E42">A31</font></i>: <font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c12">n</font> = 
<font color="Maroon" title="c9">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="txt" href="amistd_1.html#E2:27_1_5"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="amistd_1.html#E2:27_1_5"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="finseq_4.html#T48" title="FINSEQ_4:th.48">FINSEQ_4:48</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c12">n</font>
<b>by </b><i><a class="txt" href="amistd_1.html#E6:27_1_5"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><a NAME="E11:27_1_5"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="amistd_1.html#E1:27_1_5"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:27_1_5"/>
<font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E6:27_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="amistd_1.html#E1:27_1_5"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="amistd_1.html#E9:27_1_5"><i><font color="Green" title="E41">A30</font></i></a>, <a class="txt" href="amistd_1.html#E10:27_1_5"><i><font color="Green" title="E42">A31</font></i></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><a NAME="E35:27_1"/><b>then </b><i><font color="Green" title="E35">A32</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E27:27_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="txt" href="amistd_1.html#E30:27_1"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="amistd_1.html#E33:27_1"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="amistd_1.html#D8" target="_self" title="AMISTD_1:def.8">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">F2</font> = <span class="p1"><a href="ami_1.html#K24" title="AMI_1:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span></span><a href="ami_1.html#K24" title="AMI_1:func.24">*&gt;</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#M5" title="AMI_1:mode.5">IL-FinSequence</a> of <font color="Maroon" title="c3">S</font> ;<br/><a NAME="E37:27_1"/><i><font color="Green" title="E36">A33</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E38:27_1"/><b>then </b><i><font color="Green" title="E37">A34</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F2</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F2</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E38">A35</font></i>: <font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> 1 = 
<font color="Maroon" title="c12">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">m</font>
<b>by </b><i><a class="txt" href="amistd_1.html#E27:27_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><i><font color="Green" title="E39">A36</font></i>: <font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">F2</font></span>)</span> = 
<font color="Maroon" title="c12">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="amistd_1.html#E37:27_1"><i><font color="Green" title="E36">A33</font></i></a>, <a class="txt" href="amistd_1.html#E38:27_1"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c6">F</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E12:27_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="amistd_1.html#E13:27_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">F2</font> implies <font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c13">n</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:27_1_8"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">F2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c13">n</font></span>)</span></span><br/><a NAME="E2:27_1_8"/><b>then </b>
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:27_1_8"/><b>then </b><i><font color="Green" title="E40">A37</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/><b>then </b><i><font color="Green" title="E41">A38</font></i>: <font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c12">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="amistd_1.html#E38:27_1"><i><font color="Green" title="E37">A34</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E18:27_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c13">n</font> = 
<font color="Maroon" title="c12">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="amistd_1.html#E38:27_1"><i><font color="Green" title="E37">A34</font></i></a>, <a class="txt" href="amistd_1.html#E3:27_1_8"><i><font color="Green" title="E40">A37</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><a NAME="E6:27_1_8"/>
<font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E6:27_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="amistd_1.html#E4:27_1_8"><i><font color="Green" title="E41">A38</font></i></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><a NAME="E42:27_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c11">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E39:27_1"><i><font color="Green" title="E38">A35</font></i></a>, <a class="txt" href="amistd_1.html#E40:27_1"><i><font color="Green" title="E39">A36</font></i></a>, <a class="ref" href="amistd_1.html#D8" target="_self" title="AMISTD_1:def.8">Def8</a></i>;<br/><a NAME="E43:27_1"/><b>then </b>
( <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> &amp; <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:27_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="amistd_1.html#E35:27_1"><i><font color="Green" title="E35">A32</font></i></a></i>;<br/><a NAME="E44:27_1"/><b>then </b><i><font color="Green" title="E40">A39</font></i>: 
( <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> or <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_9"/><i><font color="Green" title="E41">A40</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_1_9"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">F1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E11:27_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="finseq_4.html#T50" title="FINSEQ_4:th.50">FINSEQ_4:50</a></i>;<br/><a NAME="E3:27_1_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">F1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:27_1_9"/><b>then </b><i><font color="Green" title="E42">A41</font></i>: 
(  not <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">F1</font> or  not <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:27_1_9"/><i><font color="Green" title="E43">A42</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F1</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:27_1_9"/><b>then </b>
<font color="Maroon" title="c9">F1</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F1</font></span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>hence </b><a NAME="E7:27_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="amistd_1.html#E27:27_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="txt" href="amistd_1.html#E33:27_1"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="amistd_1.html#E1:27_1_9"><i><font color="Green" title="E41">A40</font></i></a>, <a class="txt" href="amistd_1.html#E4:27_1_9"><i><font color="Green" title="E42">A41</font></i></a>, <a class="txt" href="amistd_1.html#E5:27_1_9"><i><font color="Green" title="E43">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E46:27_1"/>
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E6:27_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="amistd_1.html#E20:27_1"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="amistd_1.html#E21:27_1"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="amistd_1.html#E27:27_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="txt" href="amistd_1.html#E44:27_1"><i><font color="Green" title="E40">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font></span>)</span> holds <br/><font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font></span><br/><b>let </b><font color="Maroon" title="c13">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font></span>)</span> implies <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> )</span><br/><b>assume </b><a NAME="E47:27_1"/><i><font color="Green" title="E41">A43</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font></span><br/><b>reconsider </b><font color="Maroon" title="c14">F</font> = <span class="p1"><a href="ami_1.html#K24" title="AMI_1:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span><a href="ami_1.html#K24" title="AMI_1:func.24">*&gt;</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#M5" title="AMI_1:mode.5">IL-FinSequence</a> of <font color="Maroon" title="c3">S</font> ;<br/><a NAME="E49:27_1"/><i><font color="Green" title="E42">A44</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E50:27_1"/><b>then </b><i><font color="Green" title="E43">A45</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">F</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">F</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E44">A46</font></i>: <font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> 1 = 
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><i><font color="Green" title="E45">A47</font></i>: <font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font></span>)</span> = 
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="amistd_1.html#E49:27_1"><i><font color="Green" title="E42">A44</font></i></a>, <a class="txt" href="amistd_1.html#E50:27_1"><i><font color="Green" title="E43">A45</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c13">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> &amp; <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font> implies <font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c15">n</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:27_1_12"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> &amp; <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c15">n</font></span>)</span></span><br/><a NAME="E2:27_1_12"/><b>then </b>
( <font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:27_1_12"/><b>then </b><i><font color="Green" title="E46">A48</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/><b>then </b><i><font color="Green" title="E47">A49</font></i>: <font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="amistd_1.html#E50:27_1"><i><font color="Green" title="E43">A45</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c13">j</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c15">n</font> = 
<font color="Maroon" title="c14">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="amistd_1.html#E50:27_1"><i><font color="Green" title="E43">A45</font></i></a>, <a class="txt" href="amistd_1.html#E3:27_1_12"><i><font color="Green" title="E46">A48</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><a NAME="E6:27_1_12"/>
<font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E47:27_1"><i><font color="Green" title="E41">A43</font></i></a>, <a class="txt" href="amistd_1.html#E4:27_1_12"><i><font color="Green" title="E47">A49</font></i></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><a NAME="E54:27_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">k</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E51:27_1"><i><font color="Green" title="E44">A46</font></i></a>, <a class="txt" href="amistd_1.html#E52:27_1"><i><font color="Green" title="E45">A47</font></i></a>, <a class="ref" href="amistd_1.html#D8" target="_self" title="AMISTD_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E55:27_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:27_1"><i><font color="Green" title="E13">A2</font></i></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>assume </b><a NAME="E3:27"/><i><font color="Green" title="E13">A50</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">k</font></span>)</span> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> iff <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">n</font> )</span><br/>

<b>let </b><font color="Maroon" title="c5">m</font>, <font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> iff <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font> implies <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:27_2"/><i><font color="Green" title="E14">A51</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1"/>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> )
 </span><b>by </b><i><a class="txt" href="amistd_1.html#E1:27_2"><i><font color="Green" title="E14">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_1"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><b>hence </b><a NAME="E2:27_2_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font>
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_2"/><i><font color="Green" title="E15">A52</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font></span><br/><div class="add"><b>thus </b><a NAME="E2:27_2_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c7">mn</font> = <font color="Maroon" title="c6">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">m</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">f'</font> = <font color="Maroon" title="c4">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">IL</font> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">IL</font> = <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c9">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">IL</font><b> such that </b><br/><a NAME="E3:27_2_1_2_1"/><i><font color="Green" title="E16">A53</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E4:27_2_1_2_1"/><i><font color="Green" title="E17">A54</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F</font> holds <br/><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c10">F</font> = <font color="Maroon" title="c9">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#M5" title="AMI_1:mode.5">IL-FinSequence</a> of <font color="Maroon" title="c3">S</font> <b>by </b><i><a class="txt" href="amistd_1.html#E3:27_2_1_2_1"><i><font color="Green" title="E16">A53</font></i></a>, <a class="ref" href="ami_1.html#D34" title="AMI_1:def.34">AMI_1:def 34</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">F</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="amistd_1.html#D8" target="_self" title="AMISTD_1:def.8">AMISTD_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Olive" title="b1">n</font></span>)</span> ) )</span><br/>

<a NAME="E6:27_2_1_2_1"/>
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="amistd_1.html#E1:27_2_1_2"><i><font color="Green" title="E15">A52</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<a NAME="E7:27_2_1_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E8:27_2_1_2_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:27_2_1_2_1"/><b>then </b><i><font color="Green" title="E18">A55</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E10:27_2_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E11:27_2_1_2_1"/><b>then </b><i><font color="Green" title="E19">A56</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F</font> )
 
<b>by </b><i><a class="txt" href="amistd_1.html#E3:27_2_1_2_1"><i><font color="Green" title="E16">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> 1 = 
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="amistd_1.html#E11:27_2_1_2_1"><i><font color="Green" title="E19">A56</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E4:27_2_1_2_1"><i><font color="Green" title="E17">A54</font></i></a>, <a class="txt" href="amistd_1.html#E11:27_2_1_2_1"><i><font color="Green" title="E19">A56</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Olive" title="b1">n</font></span>)</span> ) )</span><br/>

<b>thus </b><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font></span>)</span> = 
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font></span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E11:27_2_1_2_1"><i><font color="Green" title="E19">A56</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E3:27_2_1_2_1"><i><font color="Green" title="E16">A53</font></i></a>, <a class="txt" href="amistd_1.html#E4:27_2_1_2_1"><i><font color="Green" title="E17">A54</font></i></a>, <a class="txt" href="amistd_1.html#E11:27_2_1_2_1"><i><font color="Green" title="E19">A56</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="txt" href="amistd_1.html#E9:27_2_1_2_1"><i><font color="Green" title="E18">A55</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Olive" title="b1">n</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c11">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> implies <font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c11">p</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E14:27_2_1_2_1"/><i><font color="Green" title="E20">A58</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c11">p</font></span>)</span></span><br/>

<a NAME="E15:27_2_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E16:27_2_1_2_1"/><b>then </b><i><font color="Green" title="E21">A59</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F</font> &amp; <font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F</font> )
 
<b>by </b><i><a class="txt" href="amistd_1.html#E14:27_2_1_2_1"><i><font color="Green" title="E20">A58</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:27_2_1_2_1"/><i><font color="Green" title="E22">A61</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<i><font color="Green" title="E23">A62</font></i>: <font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E16:27_2_1_2_1"><i><font color="Green" title="E21">A59</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E4:27_2_1_2_1"><i><font color="Green" title="E17">A54</font></i></a>, <a class="txt" href="amistd_1.html#E16:27_2_1_2_1"><i><font color="Green" title="E21">A59</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E14:27_2_1_2_1"><i><font color="Green" title="E20">A58</font></i></a>, <a class="txt" href="amistd_1.html#E17:27_2_1_2_1"><i><font color="Green" title="E22">A61</font></i></a>, <a class="ref" href="binarith.html#T56" title="BINARITH:th.56">BINARITH:56</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
;<br/>
<font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c11">p</font> = 
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font>
<b>by </b><i><a class="txt" href="amistd_1.html#E16:27_2_1_2_1"><i><font color="Green" title="E21">A59</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="amistd_1.html#E4:27_2_1_2_1"><i><font color="Green" title="E17">A54</font></i></a>, <a class="txt" href="amistd_1.html#E16:27_2_1_2_1"><i><font color="Green" title="E21">A59</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E20:27_2_1_2_1"/>
<font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c11">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E3:27"><i><font color="Green" title="E13">A50</font></i></a>, <a class="txt" href="amistd_1.html#E18:27_2_1_2_1"><i><font color="Green" title="E23">A62</font></i></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></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:27"/><i><font color="Green" title="E14">A63</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font></span><br/>

<a NAME="E6:27"/><i><font color="Green" title="E15">A64</font></i>: 
( <font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="txt" href="amistd_1.html#E1:27"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E7:27"/><i><font color="Green" title="E16">A65</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">F</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#M5" title="AMI_1:mode.5">IL-FinSequence</a> of <font color="Maroon" title="c3">S</font><b> such that </b><br/><a NAME="E9:27"/><i><font color="Green" title="E17">A66</font></i>: 
( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c6">n</font> )
 <b>and </b><br/><a NAME="E10:27"/><i><font color="Green" title="E18">A67</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> holds <br/><font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E5:27"><i><font color="Green" title="E14">A63</font></i></a>, <a class="ref" href="amistd_1.html#D8" target="_self" title="AMISTD_1:def.8">Def8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> implies  ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> ) );<br/>
<a NAME="E11:27"/><i><font color="Green" title="E19">A68</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E20">A69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:27_3"/><i><font color="Green" title="E21">A70</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1"/><i><font color="Green" title="E22">A71</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1"><b>suppose </b></a><a NAME="E1:27_3_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:27_3_1_1_1"/>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> )
 <b>by </b><i><a class="txt" href="amistd_1.html#E9:27"><i><font color="Green" title="E17">A66</font></i></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_2"><b>suppose </b></a><a NAME="E1:27_3_1_1_2"/><i><font color="Green" title="E23">A72</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> )</span><br/><div class="add"><a NAME="E2:27_3_1_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:27_3_1"><i><font color="Green" title="E22">A71</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_3_1_1_2"/><b>then </b><i><font color="Green" title="E24">A73</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E10:27"><i><font color="Green" title="E18">A67</font></i></a>, <a class="txt" href="amistd_1.html#E1:27_3_1_1_2"><i><font color="Green" title="E23">A72</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:27_3_1_1_2"/><i><font color="Green" title="E25">A74</font></i>: 
( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font> )
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:27_3"><i><font color="Green" title="E21">A70</font></i></a>, <a class="txt" href="amistd_1.html#E1:27_3_1"><i><font color="Green" title="E22">A71</font></i></a>, <a class="txt" href="amistd_1.html#E1:27_3_1_1_2"><i><font color="Green" title="E23">A72</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E6:27_3_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">IL</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E6:27"><i><font color="Green" title="E15">A64</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">l1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:27_3_1_1_2"/><i><font color="Green" title="E26">A75</font></i>: 
( <font color="Maroon" title="c10">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">l1</font> = <font color="Maroon" title="c10">l1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="amistd_1.html#E8:27_3_1_1_2"><i><font color="Green" title="E26">A75</font></i></a></i>;<br/><a NAME="E10:27_3_1_1_2"/>
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">l1</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E3:27"><i><font color="Green" title="E13">A50</font></i></a>, <a class="txt" href="amistd_1.html#E3:27_3_1_1_2"><i><font color="Green" title="E24">A73</font></i></a>, <a class="txt" href="amistd_1.html#E5:27_3_1_1_2"><i><font color="Green" title="E25">A74</font></i></a>, <a class="txt" href="amistd_1.html#E8:27_3_1_1_2"><i><font color="Green" title="E26">A75</font></i></a></i>;<br/><b>hence </b><a NAME="E11:27_3_1_1_2"/>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> )
 <b>by </b><i><a class="txt" href="amistd_1.html#E5:27_3_1_1_2"><i><font color="Green" title="E25">A74</font></i></a>, <a class="txt" href="amistd_1.html#E8:27_3_1_1_2"><i><font color="Green" title="E26">A75</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:27_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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>
<a NAME="E13:27"/><i><font color="Green" title="E21">A76</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="amistd_1.html#E11:27"><i><font color="Green" title="E19">A68</font></i></a>, <a class="txt" href="amistd_1.html#E12:27"><i><font color="Green" title="E20">A69</font></i></a>);<br/></i>
<a NAME="E14:27"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E15:27"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c7">F</font> <a href="ami_1.html#K22" title="AMI_1:func.22">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b1">l</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> )
 
<b>by </b><i><a class="txt" href="amistd_1.html#E13:27"><i><font color="Green" title="E21">A76</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:27"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E6:27"><i><font color="Green" title="E15">A64</font></i></a>, <a class="txt" href="amistd_1.html#E7:27"><i><font color="Green" title="E16">A65</font></i></a>, <a class="txt" href="amistd_1.html#E9:27"><i><font color="Green" title="E17">A66</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</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>
