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<span class="kw">let </span><font color="Maroon" title="c1">H1</font>, <font color="Maroon" title="c2">H2</font> be   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</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">S</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">R</font> being  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Olive" title="b1">S</font>,<font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b4">BASSIGN</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Olive" title="b1">S</font></span>)</span><br/> for <font color="Olive" title="b5">kai</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a> ,the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">BasicAssign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Olive" title="b2">R</font>,<font color="Olive" title="b4">BASSIGN</font></span>)</span> holds <br/> ( <font color="Olive" title="b3">s</font>,<font color="Olive" title="b5">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff  ex <font color="Olive" title="b6">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s</font> &amp;  ex <font color="Olive" title="b7">m</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 <br/>( ( for <font color="Olive" title="b8">j</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 <font color="Olive" title="b8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">m</font> holds <br/><font color="Olive" title="b6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">j</font>,<font color="Olive" title="b5">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b6">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">m</font>,<font color="Olive" title="b5">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c3">S</font><br/> for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font><br/> for <font color="Olive" title="b3">BASSIGN</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c3">S</font></span>)</span><br/> for <font color="Olive" title="b4">kai</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a> ,the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">BasicAssign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b3">BASSIGN</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">s</font>,<font color="Olive" title="b4">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff  ex <font color="Olive" title="b5">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b5">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> &amp;  ex <font color="Olive" title="b6">m</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 <br/>( ( for <font color="Olive" title="b7">j</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 <font color="Olive" title="b7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">m</font> holds <br/><font color="Olive" title="b5">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">j</font>,<font color="Olive" title="b4">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b5">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">m</font>,<font color="Olive" title="b4">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">R</font> be   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font><br/> for <font color="Olive" title="b2">BASSIGN</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c3">S</font></span>)</span><br/> for <font color="Olive" title="b3">kai</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a> ,the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">BasicAssign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c4">R</font>,<font color="Olive" title="b2">BASSIGN</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">s</font>,<font color="Olive" title="b3">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff  ex <font color="Olive" title="b4">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b4">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp;  ex <font color="Olive" title="b5">m</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 <br/>( ( for <font color="Olive" title="b6">j</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 <font color="Olive" title="b6">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">m</font> holds <br/><font color="Olive" title="b4">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">j</font>,<font color="Olive" title="b3">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b4">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">m</font>,<font color="Olive" title="b3">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">BASSIGN</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c3">S</font></span>)</span><br/> for <font color="Olive" title="b2">kai</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a> ,the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">BasicAssign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c4">R</font>,<font color="Olive" title="b1">BASSIGN</font></span>)</span> holds <br/> ( <font color="Maroon" title="c5">s</font>,<font color="Olive" title="b2">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff  ex <font color="Olive" title="b3">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b3">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b4">m</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 <br/>( ( for <font color="Olive" title="b5">j</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 <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b3">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">j</font>,<font color="Olive" title="b2">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b3">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">m</font>,<font color="Olive" title="b2">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">BASSIGN</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c3">S</font></span>)</span>; <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">kai</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a> ,the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">BasicAssign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c6">BASSIGN</font></span>)</span> holds <br/> ( <font color="Maroon" title="c5">s</font>,<font color="Olive" title="b1">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff  ex <font color="Olive" title="b2">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b2">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b3">m</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 <br/>( ( for <font color="Olive" title="b4">j</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 <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b2">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">j</font>,<font color="Olive" title="b1">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b2">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">m</font>,<font color="Olive" title="b1">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">kai</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a> ,the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">BasicAssign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c6">BASSIGN</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b2">m</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 <br/>( ( for <font color="Olive" title="b3">j</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 <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) )</span><br/>













<a NAME="E1:139"/><span class="lab"><font color="Green" title="E105">A1</font></span>: 
(  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b2">m</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 <br/>( ( for <font color="Olive" title="b3">j</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 <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font> ) &amp; <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font> ) ) implies  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b2">m</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 <br/>( ( for <font color="Olive" title="b3">j</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 <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c8">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font><span class="kw"> such that </span><a NAME="E2:139_1"/><span class="lab"><font color="Green" title="E106">A2</font></span>: 
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">and </span><br/><a NAME="E3:139_1"/><span class="lab"><font color="Green" title="E107">A3</font></span>: 
 ex <font color="Olive" title="b1">m</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 <br/>( ( for <font color="Olive" title="b2">j</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 <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font> ) &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b2">m</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 <br/>( ( for <font color="Olive" title="b3">j</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 <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">pai</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b1">m</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 <br/>( ( for <font color="Olive" title="b2">j</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 <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">m</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> <span class="kw"> such that </span><br/><a NAME="E5:139_1"/><span class="lab"><font color="Green" title="E108">A4</font></span>: 
for <font color="Olive" title="b1">j</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 <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font>
 <span class="kw">and </span><br/><a NAME="E6:139_1"/><span class="lab"><font color="Green" title="E109">A5</font></span>: 
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:139_1"><span class="lab"><font color="Green" title="E107">A3</font></span></a></span>;<br/>
<a NAME="E7:139_1"/><span class="lab"><font color="Green" title="E110">A6</font></span>: 
for <font color="Olive" title="b1">j</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 <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">j</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"> ( <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> implies <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:139_1_1"/>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font></span><br/>

<a NAME="E2:139_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:139_1"><span class="lab"><font color="Green" title="E108">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:139_1_1"/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D60" target="_self" title="MODELC_1:def.60">Def60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:139_1"/>
( <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font> iff <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D60" target="_self" title="MODELC_1:def.60">Def60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:139_1"/>
( <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b1">m</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 <br/>( ( for <font color="Olive" title="b2">j</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 <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:139_1"><span class="lab"><font color="Green" title="E106">A2</font></span></a>, <a class="txt" href="#E6:139_1"><span class="lab"><font color="Green" title="E109">A5</font></span></a>, <a class="txt" href="#E7:139_1"><span class="lab"><font color="Green" title="E110">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:139"/><span class="lab"><font color="Green" title="E106">A7</font></span>: 
(  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b2">m</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 <br/>( ( for <font color="Olive" title="b3">j</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 <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) implies  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b2">m</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 <br/>( ( for <font color="Olive" title="b3">j</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 <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font> ) &amp; <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c8">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font><span class="kw"> such that </span><a NAME="E2:139_2"/><span class="lab"><font color="Green" title="E107">A8</font></span>: 
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">and </span><br/><a NAME="E3:139_2"/><span class="lab"><font color="Green" title="E108">A9</font></span>: 
 ex <font color="Olive" title="b1">m</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 <br/>( ( for <font color="Olive" title="b2">j</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 <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b2">m</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 <br/>( ( for <font color="Olive" title="b3">j</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 <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font> ) &amp; <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">pai</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b1">m</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 <br/>( ( for <font color="Olive" title="b2">j</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 <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font> ) &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">m</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> <span class="kw"> such that </span><br/><a NAME="E5:139_2"/><span class="lab"><font color="Green" title="E109">A10</font></span>: 
for <font color="Olive" title="b1">j</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 <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font>
 <span class="kw">and </span><br/><a NAME="E6:139_2"/><span class="lab"><font color="Green" title="E110">A11</font></span>: 
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:139_2"><span class="lab"><font color="Green" title="E108">A9</font></span></a></span>;<br/>
<a NAME="E7:139_2"/><span class="lab"><font color="Green" title="E111">A12</font></span>: 
for <font color="Olive" title="b1">j</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 <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">j</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"> ( <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font> implies <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:139_2_1"/>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font></span><br/>

<a NAME="E2:139_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:139_2"><span class="lab"><font color="Green" title="E109">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:139_2_1"/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D60" target="_self" title="MODELC_1:def.60">Def60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:139_2"/>
( <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font> iff <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D60" target="_self" title="MODELC_1:def.60">Def60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:139_2"/>
( <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b1">m</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 <br/>( ( for <font color="Olive" title="b2">j</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 <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font> ) &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:139_2"><span class="lab"><font color="Green" title="E107">A8</font></span></a>, <a class="txt" href="#E6:139_2"><span class="lab"><font color="Green" title="E110">A11</font></span></a>, <a class="txt" href="#E7:139_2"><span class="lab"><font color="Green" title="E111">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:139"/>
( <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff <font color="Maroon" title="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font></span>)</span>,<font color="Maroon" title="c7">kai</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D60" target="_self" title="MODELC_1:def.60">Def60</a></span>;<br/>
<a NAME="E4:139"/><span class="kw">then </span>
( <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff <font color="Maroon" title="c5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c1">H1</font>,<font color="Maroon" title="c7">kai</font></span>)</span> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K24" title="MODELC_1:func.24">Evaluate</a> <font color="Maroon" title="c2">H2</font>,<font color="Maroon" title="c7">kai</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T9" target="_self" title="MODELC_1:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:139"/>
( <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> <a href="modelc_1.html#K12" title="MODELC_1:func.12">EU</a> <font color="Maroon" title="c2">H2</font> iff  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c4">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> &amp;  ex <font color="Olive" title="b2">m</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 <br/>( ( for <font color="Olive" title="b3">j</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 <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c1">H1</font> ) &amp; <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">m</font>,<font color="Maroon" title="c7">kai</font> <a href="modelc_1.html#R4" title="MODELC_1:pred.4">|=</a> <font color="Maroon" title="c2">H2</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:139"><span class="lab"><font color="Green" title="E105">A1</font></span></a>, <a class="txt" href="#E2:139"><span class="lab"><font color="Green" title="E106">A7</font></span></a>, <a class="ref" href="modelc_1.html#T16" target="_self" title="MODELC_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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