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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b3">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">num</font> being   <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>,<span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Olive" title="b2">S</font></span>)</span>  st <font color="Olive" title="b3">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Olive" title="b2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">D</font> &amp;  not <font color="Olive" title="b1">X</font> is <font color="Olive" title="b3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/>( <span class="p1">(<span class="default">(<font color="Olive" title="b3">D</font>,<font color="Olive" title="b4">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Olive" title="b3">D</font>,<font color="Olive" title="b4">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Olive" title="b3">D</font>,<font color="Olive" title="b4">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> is <font color="Olive" title="b3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">num</font> being   <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>,<span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Olive" title="b1">S</font></span>)</span>  st <font color="Olive" title="b2">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> &amp;  not <font color="Maroon" title="c3">X</font> is <font color="Olive" title="b2">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/>( <span class="p1">(<span class="default">(<font color="Olive" title="b2">D</font>,<font color="Olive" title="b3">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Olive" title="b2">D</font>,<font color="Olive" title="b3">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Olive" title="b2">D</font>,<font color="Olive" title="b3">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> is <font color="Olive" title="b2">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">S</font> be   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c4">S</font><br/>  for <font color="Olive" title="b2">num</font> being   <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>,<span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c4">S</font></span>)</span>  st <font color="Olive" title="b1">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp;  not <font color="Maroon" title="c3">X</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/>( <span class="p1">(<span class="default">(<font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">D</font> be   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">num</font> being   <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>,<span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c4">S</font></span>)</span>  st <font color="Maroon" title="c5">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp;  not <font color="Maroon" title="c3">X</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/>( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Olive" title="b1">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Olive" title="b1">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Olive" title="b1">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">FF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c4">S</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">num</font> be   <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>,<span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c4">S</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp;  not <font color="Maroon" title="c3">X</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  implies ( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">f</font> = (<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font>;<br/>
<span class="kw">assume </span><a NAME="E1:247"/><span class="lab"><font color="Green" title="E134">B10</font></span>: 
( <font color="Maroon" title="c5">D</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  or ( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:247"/><span class="lab"><font color="Green" title="E135">B11</font></span>: 
 not <font color="Maroon" title="c3">X</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  );<br/>
<a NAME="E3:247"/><span class="lab"><font color="Green" title="E136">B0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:247"><span class="lab"><font color="Green" title="E135">B11</font></span></a>, <a class="ref" href="fomodel4.html#D74" target="_self" title="FOMODEL4:def.74">DefAdd2</a></span>;<br/>
<a NAME="E4:247"/><span class="lab"><font color="Green" title="E137">B1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="247_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">fkn1</font> = (<font color="Maroon" title="c5">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">num</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) <a href="fomodel4.html#K47" title="FOMODEL4:func.47">AddFormulaTo</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">fkn</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:247_1"/><span class="lab"><font color="Green" title="E138">C1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<a NAME="E2:247_1"/><span class="lab"><font color="Green" title="E139">CC0</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">num</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) <a href="fomodel4.html#K47" title="FOMODEL4:func.47">AddFormulaTo</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>

<br/>.= 
(<font color="Maroon" title="c5">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">num</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>) <a href="fomodel4.html#K47" title="FOMODEL4:func.47">AddFormulaTo</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D74" target="_self" title="FOMODEL4:def.74">DefAdd2</a></span>
;<br/>
<a NAME="E4:247_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:247_1"><span class="lab"><font color="Green" title="E139">CC0</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:247_1"/>
<span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:247_1"><span class="lab"><font color="Green" title="E138">C1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">phii</font> = <font color="Maroon" title="c7">num</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c4">S</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">phi</font> = <font color="Maroon" title="c12">phii</font> as   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c4">S</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">psi</font> =  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c13">phi</font> as   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c4">S</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">psii</font> = <font color="Maroon" title="c14">psi</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c4">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T16" title="FOMODEL2:th.16">FOMODEL2:16</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">fn</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">fN</font> = (<font color="Maroon" title="c5">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">num</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="fomodel4.html#K47" title="FOMODEL4:func.47">AddFormulaTo</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[] <span class="kw">means </span> <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c12">phii</font> is <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> ;<br/>
<a NAME="E10:247_1"/><span class="lab"><font color="Green" title="E140">C2</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c5">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">num</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="fomodel4.html#K47" title="FOMODEL4:func.47">AddFormulaTo</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D74" target="_self" title="FOMODEL4:def.74">DefAdd2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="247_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:247_1_2"/>
(  not <font color="Maroon">S<sub>2</sub></font>[] or <font color="Maroon">S<sub>2</sub></font>[] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="247_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:247_1_2_1"/><span class="lab"><font color="Green" title="E141">D0</font></span>: 
 not <font color="Maroon">S<sub>2</sub></font>[]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/><div class="add"><a NAME="E2:247_1_2_1"/><span class="kw">then </span>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">phii</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:247"><span class="lab"><font color="Green" title="E134">B10</font></span></a>, <a class="txt" href="fomodel4.html#E135"><span class="lab"><font color="Green" title="E105">Lm2b</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:247_1_2_1"/>
 not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:247_1"><span class="lab"><font color="Green" title="E140">C2</font></span></a>, <a class="txt" href="#E1:247_1_2_1"><span class="lab"><font color="Green" title="E141">D0</font></span></a>, <a class="ref" href="fomodel4.html#D73" target="_self" title="FOMODEL4:def.73">DefAdd1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="247_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:247_1_2_2"/><span class="lab"><font color="Green" title="E141">D0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/><div class="add"><a NAME="E2:247_1_2_2"/><span class="kw">then </span>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c12">phii</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E138"><span class="lab"><font color="Green" title="E108">Lm3</font></span></a>, <a class="txt" href="#E1:247"><span class="lab"><font color="Green" title="E134">B10</font></span></a>, <a class="txt" href="#E1:247_1"><span class="lab"><font color="Green" title="E138">C1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:247_1_2_2"/>
 not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:247_1"><span class="lab"><font color="Green" title="E140">C2</font></span></a>, <a class="txt" href="#E1:247_1_2_2"><span class="lab"><font color="Green" title="E141">D0</font></span></a>, <a class="ref" href="fomodel4.html#D73" target="_self" title="FOMODEL4:def.73">DefAdd1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:247"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:247"><span class="lab"><font color="Green" title="E136">B0</font></span></a>, <a class="txt" href="#E4:247"><span class="lab"><font color="Green" title="E137">B1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:247"/>
( <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  not <span class="p1">(<span class="default">(<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">num</font>) <a href="fomodel4.html#K48" title="FOMODEL4:func.48">AddFormulasTo</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> is <font color="Maroon" title="c5">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
