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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Olive" title="b1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">R</font> holds <br/> <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<span class="p1">(<span class="default"><a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Olive" title="b1">R</font>,<font color="Maroon" title="c1">m</font>)</span>)</span>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Olive" title="b1">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">R</font> implies  <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<span class="p1">(<span class="default"><a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c1">m</font>)</span>)</span>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>) )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<span class="p1">(<span class="default"><a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c1">m</font>)</span>)</span>,$1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span>);<br/>
<span class="kw">assume </span><a NAME="E1:112"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">R</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<span class="p1">(<span class="default"><a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c1">m</font>)</span>)</span>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span><br/>

<a NAME="E2:112"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T74" target="_self" onmouseover="rs('funct_7/T74')" onmouseout="rh()">Th73</a></span>;<br/>
<a NAME="E3:112"/><span class="kw">then </span>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="funct_7.html#T74" target="_self" onmouseover="rs('funct_7/T74')" onmouseout="rh()">Th73</a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/>
<span class="kw">then </span> <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<span class="p1">(<span class="default"><a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c1">m</font>)</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) = 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T68" target="_self" onmouseover="rs('funct_7/T68')" onmouseout="rh()">Th67</a></span>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <font color="Maroon" title="c3">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>
<br/>.= 
 <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T68" target="_self" onmouseover="rs('funct_7/T68')" onmouseout="rh()">Th67</a></span>
;<br/>
<a NAME="E5:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E6:112"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="funct_7.html#E93"><span class="lab"><font color="Green" title="E56">Lm5</font></span></a></span>;<br/>
<a NAME="E7:112"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E5:112"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E6:112"><span class="lab"><font color="Green" title="E60">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:112"/>
 <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<span class="p1">(<span class="default"><a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c1">m</font>)</span>)</span>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K8" title="FUNCT_7:func.8">iter</a> (<font color="Maroon" title="c3">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">n</font></span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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