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<div>:: <span class="kw">deftheorem </span><span class="lab"><font color="Green" title="E3">Def3</font></span>   defines <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <a onclick="hs(this)" href="javascript:()">INTPRO_2:def 3 : <br/></a><span> for <font color="Olive" title="b1">PR</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>,<a href="intpro_2.html#K2" title="INTPRO_2:func.2">Proof_Step_Kinds_IPC</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a> holds <br/> ( ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">X</font> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b4">p</font></span>)</span> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b6">r</font></span>)</span></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b6">r</font></span>)</span></span>)</span> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b4">p</font> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b5">q</font> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Olive" title="b5">q</font></span>)</span></span>)</span> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Olive" title="b5">q</font></span>)</span> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Olive" title="b5">q</font></span>)</span> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 8 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b6">r</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b6">r</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b4">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b6">r</font></span>)</span></span>)</span> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="intpro_1.html#K2" title="INTPRO_1:func.2">FALSUM</a> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b4">p</font> ) ) &amp; ( <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 10 implies ( <font color="Olive" title="b1">PR</font>,<font color="Olive" title="b2">n</font> <a href="intpro_2.html#R1" title="INTPRO_2:pred.1">is_a_correct_step_wrt_IPC</a> <font color="Olive" title="b3">X</font> iff  ex <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">p</font>, <font color="Olive" title="b7">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">j</font> &amp; <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">i</font> &amp; <font color="Olive" title="b6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <font color="Olive" title="b7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">PR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Olive" title="b7">q</font> ) ) ) );<br/></span></div>
