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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">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">E</font> <a href="polnot_1.html#K14" title="POLNOT_1:func.14">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> for <font color="Olive" title="b3">F</font> being   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">E</font>  st  <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> holds <br/> ex <font color="Olive" title="b4">G</font>, <font color="Olive" title="b5">H</font> being   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">E</font> st <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b4">G</font>,<font color="Olive" title="b5">H</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">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">E</font> <a href="polnot_1.html#K14" title="POLNOT_1:func.14">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> for <font color="Olive" title="b2">F</font> being   <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>  st  <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> holds <br/> ex <font color="Olive" title="b3">G</font>, <font color="Olive" title="b4">H</font> being   <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> st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">G</font>,<font color="Olive" title="b4">H</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c2">E</font> <a href="polnot_1.html#K14" title="POLNOT_1:func.14">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies  for <font color="Olive" title="b1">F</font> being   <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>  st  <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> holds <br/> ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <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> st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font>) )</span><br/>





<span class="kw">assume </span><a NAME="E1:171"/><span class="lab"><font color="Green" title="E94">A1</font></span>: 
<font color="Maroon" title="c2">E</font> <a href="polnot_1.html#K14" title="POLNOT_1:func.14">.</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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">F</font> being   <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>  st  <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> holds <br/> ex <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being   <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> st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> implies  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <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> st <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:171"/><span class="lab"><font color="Green" title="E95">A2</font></span>: 
 <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</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">  ex <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being   <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> st <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">W</font> =  <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>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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 href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> 2<span class="kw"> such that </span><br/><a NAME="E4:171"/><span class="lab"><font color="Green" title="E96">A3</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K22" title="POLNOT_1:func.22">Polish-operation</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:171"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E2:171"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="polnot_1.html#T79" target="_self" onmouseover="rs('polnot_1/T79')" onmouseout="rh()">Th79</a></span>;<br/>
<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 href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> 2 = 
<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 href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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 href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> 1</span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T6" target="_self" onmouseover="rs('polnot_1/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<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 href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">G</font>, <font color="Maroon" title="c8">H</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E7:171"/><span class="lab"><font color="Green" title="E97">A5</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">H</font>
 <span class="kw">and </span><br/><a NAME="E8:171"/><span class="lab"><font color="Green" title="E98">A6</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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 class="kw">and </span><br/><a NAME="E9:171"/><span class="lab"><font color="Green" title="E99">A7</font></span>: 
<font color="Maroon" title="c8">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">G</font> = <font color="Maroon" title="c7">G</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:171"><span class="lab"><font color="Green" title="E98">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c8">H</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:171"><span class="lab"><font color="Green" title="E99">A7</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">G</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">  ex <font color="Olive" title="b1">H</font> being   <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> st <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">G</font>,<font color="Olive" title="b1">H</font>)</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">H</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="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c10">H</font>)</span><br/>

<span class="kw">thus </span><a NAME="E12:171"/>
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K25" title="POLNOT_1:func.25">Polish-binOp</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">t</font>)</span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">G</font>,<font color="Maroon" title="c10">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:171"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E4:171"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E7:171"><span class="lab"><font color="Green" title="E97">A5</font></span></a>, <a class="ref" href="polnot_1.html#D28" target="_self" onmouseover="rs('polnot_1/D28')" onmouseout="rh()">Def28</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/>


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