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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="polnot_1.html#NM3" title="POLNOT_1:NM.3">Polish-language</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="polnot_1.html#M1" title="POLNOT_1:mode.1">Polish-arity-function</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">g</font> being   <a href="polnot_1.html#NM7" title="POLNOT_1:NM.7">Polish-recursion-function</a> of <font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">D</font><br/>  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b5">J</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">E</font>)</span>)</span><br/>  for <font color="Olive" title="b6">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b5">J</font>,<font color="Olive" title="b2">D</font><br/>  for <font color="Olive" title="b7">J1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">E</font>)</span>)</span><br/>  for <font color="Olive" title="b8">H1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b7">J1</font>,<font color="Olive" title="b2">D</font><br/>  for <font color="Olive" title="b9">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">E</font>,<font color="Olive" title="b4">n</font>) &amp; <font color="Olive" title="b6">H</font> is <font color="Olive" title="b3">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b5">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b7">J1</font> &amp; <font color="Olive" title="b8">H1</font> is <font color="Olive" title="b3">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">J</font> holds <br/><font color="Olive" title="b6">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be    <a href="polnot_1.html#M1" title="POLNOT_1:mode.1">Polish-arity-function</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">g</font> being   <a href="polnot_1.html#NM7" title="POLNOT_1:NM.7">Polish-recursion-function</a> of <font color="Maroon" title="c2">E</font>,<font color="Olive" title="b1">D</font><br/>  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b4">J</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b5">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b4">J</font>,<font color="Olive" title="b1">D</font><br/>  for <font color="Olive" title="b6">J1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b7">H1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b6">J1</font>,<font color="Olive" title="b1">D</font><br/>  for <font color="Olive" title="b8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b4">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Olive" title="b3">n</font>) &amp; <font color="Olive" title="b5">H</font> is <font color="Olive" title="b2">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b4">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b6">J1</font> &amp; <font color="Olive" title="b7">H1</font> is <font color="Olive" title="b2">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">J</font> holds <br/><font color="Olive" title="b5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="polnot_1.html#NM7" title="POLNOT_1:NM.7">Polish-recursion-function</a> of <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">D</font><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">J</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b4">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b3">J</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b5">J1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b6">H1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b5">J1</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Olive" title="b2">n</font>) &amp; <font color="Olive" title="b4">H</font> is <font color="Olive" title="b1">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">J1</font> &amp; <font color="Olive" title="b6">H1</font> is <font color="Olive" title="b1">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">J</font> holds <br/><font color="Olive" title="b4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="polnot_1.html#NM7" title="POLNOT_1:NM.7">Polish-recursion-function</a> of <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">J</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b3">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b4">J1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b5">H1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b4">J1</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b3">H</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b2">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">J1</font> &amp; <font color="Olive" title="b5">H1</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">J</font> holds <br/><font color="Olive" title="b3">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">J</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b2">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">J</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b3">J1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b4">H1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b3">J1</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c5">n</font>) &amp; <font color="Olive" title="b2">H</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b1">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">J1</font> &amp; <font color="Olive" title="b4">H1</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">J</font> holds <br/><font color="Olive" title="b2">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">J</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b2">J1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b3">H1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J1</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b4">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c5">n</font>) &amp; <font color="Olive" title="b1">H</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Maroon" title="c6">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">J1</font> &amp; <font color="Olive" title="b3">H1</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> holds <br/><font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">H</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c3">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">J1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b2">H1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">J1</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b3">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c5">n</font>) &amp; <font color="Maroon" title="c7">H</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Maroon" title="c6">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">J1</font> &amp; <font color="Olive" title="b2">H1</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> holds <br/><font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">J1</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">J1</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b2">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c5">n</font>) &amp; <font color="Maroon" title="c7">H</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Maroon" title="c6">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">J1</font> &amp; <font color="Olive" title="b1">H1</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> holds <br/><font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">H1</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">J1</font>,<font color="Maroon" title="c3">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c5">n</font>) &amp; <font color="Maroon" title="c7">H</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Maroon" title="c6">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">J1</font> &amp; <font color="Maroon" title="c9">H1</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> holds <br/><font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c5">n</font>) &amp; <font color="Maroon" title="c7">H</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Maroon" title="c6">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">J1</font> &amp; <font color="Maroon" title="c9">H1</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a>  &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> implies <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )</span><br/>



















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:159"/><span class="lab"><font color="Green" title="E87">A1</font></span>: 
<font color="Maroon" title="c6">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c5">n</font>)
 <span class="kw">and </span><br/><a NAME="E2:159"/><span class="lab"><font color="Green" title="E88">A2</font></span>: 
<font color="Maroon" title="c7">H</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a> 
 <span class="kw">and </span><br/><a NAME="E3:159"/><span class="lab"><font color="Green" title="E89">A3</font></span>: 
<font color="Maroon" title="c6">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">J1</font>
 <span class="kw">and </span><br/><a NAME="E4:159"/><span class="lab"><font color="Green" title="E90">A4</font></span>: 
<font color="Maroon" title="c9">H1</font> is <font color="Maroon" title="c4">g</font> <a href="polnot_1.html#V9" title="POLNOT_1:attr.9">-recursive</a> 
 <span class="kw">and </span><br/><a NAME="E5:159"/><span class="lab"><font color="Green" title="E91">A5</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,$1) holds <br/><font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>;<br/>
<a NAME="E6:159"/><span class="lab"><font color="Green" title="E92">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="159_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>) implies <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:159_1"/><span class="lab"><font color="Green" title="E93">A11</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>) or <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">G</font> = <font color="Maroon" title="c11">a</font> as   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font> ;<br/>
<span class="kw">assume </span><a NAME="E3:159_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font></span><br/>

<a NAME="E4:159_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A13</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D9" target="_self" onmouseover="rs('polnot_1/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E5:159_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D7" target="_self" onmouseover="rs('polnot_1/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E6:159_1"/><span class="kw">then </span>
 <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T53" target="_self" onmouseover="rs('polnot_1/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<a NAME="E7:159_1"/><span class="kw">then </span>
 <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:159_1"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="ref" href="polnot_1.html#D7" target="_self" onmouseover="rs('polnot_1/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E8:159_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A15</font></span>: 
 <a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T61" target="_self" onmouseover="rs('polnot_1/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<a NAME="E9:159_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A16</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">J</font>
 
;<br/>
<a NAME="E10:159_1"/><span class="lab"><font color="Green" title="E97">A17</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">J1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:159_1"><span class="lab"><font color="Green" title="E95">A15</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> = 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <font color="Maroon" title="c12">G</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:159"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="txt" href="#E1:159_1"><span class="lab"><font color="Green" title="E93">A11</font></span></a>, <a class="txt" href="#E8:159_1"><span class="lab"><font color="Green" title="E95">A15</font></span></a>, <a class="txt" href="#E9:159_1"><span class="lab"><font color="Green" title="E96">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <font color="Maroon" title="c12">G</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">H1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

<br/>.= 
<font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:159"><span class="lab"><font color="Green" title="E89">A3</font></span></a>, <a class="txt" href="#E4:159"><span class="lab"><font color="Green" title="E90">A4</font></span></a>, <a class="txt" href="#E1:159_1"><span class="lab"><font color="Green" title="E93">A11</font></span></a>, <a class="txt" href="#E8:159_1"><span class="lab"><font color="Green" title="E95">A15</font></span></a>, <a class="txt" href="#E10:159_1"><span class="lab"><font color="Green" title="E97">A17</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:159"/><span class="lab"><font color="Green" title="E93">A20</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="159_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:159_2"/><span class="lab"><font color="Green" title="E94">A21</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</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="c12">J2</font> =  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c11">k</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">J3</font> =  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>);<br/>
<span class="kw">let </span><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) implies <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:159_2"/><span class="lab"><font color="Green" title="E95">A22</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) or <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:159_2"/><span class="lab"><font color="Green" title="E96">A23</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:159_2_1"/>
( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> or <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:159_2_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:159_2_1_1"/><span class="lab"><font color="Green" title="E97">A24</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><a NAME="E4:159_2_1_1"/>
<font color="Maroon" title="c6">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c11">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:159"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="#E3:159_2_1_1"><span class="lab"><font color="Green" title="E97">A24</font></span></a>, <a class="ref" href="polnot_1.html#T25" target="_self" onmouseover="rs('polnot_1/T25')" onmouseout="rh()">Th25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">G</font> = <font color="Maroon" title="c14">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c11">k</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:159_2"><span class="lab"><font color="Green" title="E95">A22</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> = 
<font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:159_2"><span class="lab"><font color="Green" title="E94">A21</font></span></a>, <a class="txt" href="#E2:159_2"><span class="lab"><font color="Green" title="E95">A22</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="159_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:159_2_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font></span><br/><div class="add"><a NAME="E2:159_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:159_2_1_2"/><span class="lab"><font color="Green" title="E97">A30</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><a NAME="E5:159_2_1_2"/><span class="lab"><font color="Green" title="E98">A31</font></span>: 
 <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:159"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="#E4:159_2_1_2"><span class="lab"><font color="Green" title="E97">A30</font></span></a>, <a class="ref" href="polnot_1.html#T25" target="_self" onmouseover="rs('polnot_1/T25')" onmouseout="rh()">Th25</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">F1</font> = <font color="Maroon" title="c14">a</font> as   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:159_2"><span class="lab"><font color="Green" title="E95">A22</font></span></a></span>;<br/><a NAME="E7:159_2_1_2"/><span class="lab"><font color="Green" title="E99">A32</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c16">F1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c11">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:159_2"><span class="lab"><font color="Green" title="E96">A23</font></span></a>, <a class="ref" href="polnot_1.html#T71" target="_self" onmouseover="rs('polnot_1/T71')" onmouseout="rh()">Th71</a></span>;<br/><a NAME="E8:159_2_1_2"/>
 <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c11">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T25" target="_self" onmouseover="rs('polnot_1/T25')" onmouseout="rh()">Th25</a></span>;<br/><a NAME="E9:159_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A33</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c16">F1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:159_2_1_2"><span class="lab"><font color="Green" title="E98">A31</font></span></a>, <a class="txt" href="#E7:159_2_1_2"><span class="lab"><font color="Green" title="E99">A32</font></span></a></span>;<br/><a NAME="E10:159_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A34</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c16">F1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">J1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:159"><span class="lab"><font color="Green" title="E89">A3</font></span></a></span>;<br/><a NAME="E11:159_2_1_2"/>
 for <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c16">F1</font></span>)</span> holds <br/><font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:159_2"><span class="lab"><font color="Green" title="E94">A21</font></span></a>, <a class="txt" href="#E7:159_2_1_2"><span class="lab"><font color="Green" title="E99">A32</font></span></a>, <a class="txt" href="#E9:159_2_1_2"><span class="lab"><font color="Green" title="E100">A33</font></span></a></span>;<br/><a NAME="E12:159_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A37</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c16">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c16">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:159_2_1_2"><span class="lab"><font color="Green" title="E100">A33</font></span></a>, <a class="txt" href="#E10:159_2_1_2"><span class="lab"><font color="Green" title="E101">A34</font></span></a>, <a class="ref" href="polnot_1.html#T72" target="_self" onmouseover="rs('polnot_1/T72')" onmouseout="rh()">Th72</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> = 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <font color="Maroon" title="c16">F1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c16">F1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:159"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="txt" href="#E2:159_2"><span class="lab"><font color="Green" title="E95">A22</font></span></a>, <a class="txt" href="#E9:159_2_1_2"><span class="lab"><font color="Green" title="E100">A33</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:159"><span class="lab"><font color="Green" title="E89">A3</font></span></a>, <a class="txt" href="#E4:159"><span class="lab"><font color="Green" title="E90">A4</font></span></a>, <a class="txt" href="#E2:159_2"><span class="lab"><font color="Green" title="E95">A22</font></span></a>, <a class="txt" href="#E10:159_2_1_2"><span class="lab"><font color="Green" title="E101">A34</font></span></a>, <a class="txt" href="#E12:159_2_1_2"><span class="lab"><font color="Green" title="E102">A37</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:159"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E6:159"><span class="lab"><font color="Green" title="E92">A10</font></span></a>, <a class="txt" href="#E7:159"><span class="lab"><font color="Green" title="E93">A20</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:159"/>
<font color="Maroon" title="c7">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:159"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="#E5:159"><span class="lab"><font color="Green" title="E91">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
