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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">m</font></span>)</span>,<font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">k</font></span>)</span> holds <br/> <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">m</font></span>)</span>,<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#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<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></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> implies  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">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>  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:113"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
 <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">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>  <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/>

<span class="kw">thus </span> <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <span class="p2">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" target="_self" title="FUNCT_7:th.71">Th71</a></span>
<br/>.= 
 <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="funct_7.html#T79" target="_self" title="FUNCT_7:th.79">Th79</a></span>
<br/>.= 
 <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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