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<a NAME="E1:66_1_1"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
(  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive" title="b1">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:66_1_1_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">  ex <font color="Olive" title="b1">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default">$1 <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default">$2 <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default">$1 <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">Y</font> =  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> as   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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a></span>;<br/>
<a NAME="E3:66_1_1_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font>  ex <font color="Olive" title="b3">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</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">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">z</font>]</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">X9</font> = <font color="Maroon" title="c1">X</font> as   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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">v</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c5">X9</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c5">X9</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">z</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T26" target="_self" onmouseover="rs('algstr_4/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">z</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="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c8">z</font>]</span><br/>

<span class="kw">thus </span><a NAME="E5:66_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c8">z</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>
<span class="kw">consider </span><font color="Maroon" title="c3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E5:66_1_1_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" onmouseover="rs('binop_1/S3')" onmouseout="rh()">BINOP_1:sch 3</a>(<a class="txt" href="#E3:66_1_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">f</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">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">thus </span><a NAME="E7:66_1_1_1"/>
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Maroon" title="c4">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></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="#E5:66_1_1_1"><span class="lab"><font color="Green" title="E36">A4</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="E2:66_1_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive" title="b1">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:66_1_1_2"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">  ex <font color="Olive" title="b1">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:66_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E3:66_1_1_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">f</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E5:66_1_1_2"/>
( ( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E7:66_1_1_2"/>
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></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="#E1:66_1_1_2"><span class="lab"><font color="Green" title="E35">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:66_1_1_2"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive" title="b1">f</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="txt" href="#E2:66_1_1_2"><span class="lab"><font color="Green" title="E36">A7</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><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_1_3"><span class="kw">now </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">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive" title="b1">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive" title="b2">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Olive" title="b1">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">f2</font> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( (  for <font color="Olive">b<sub>3</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  verum ) &amp; (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>3</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/> for <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b6">n</font>, <font color="Olive" title="b7">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>3</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b4">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b5">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b4">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b7">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>3</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b5">v</font>, <font color="Olive" title="b6">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b7">n</font>, <font color="Olive" title="b8">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>3</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b5">v</font>,<font color="Olive" title="b6">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b5">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b6">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b5">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b8">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b5">v</font>, <font color="Olive" title="b6">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b7">n</font>, <font color="Olive" title="b8">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>4</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b5">v</font>,<font color="Olive" title="b6">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b5">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b6">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b5">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b8">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> ) ) ) ) ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</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"> ( (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Maroon" title="c2">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Maroon" title="c3">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f2</font> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( (  for <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  verum ) &amp; (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E1:66_1_1_3"/>
(  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Maroon" title="c2">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Maroon" title="c3">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f2</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="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( (  for <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  verum ) &amp; (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:66_1_1_3_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
 not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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"> (  ex <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp;  not <font color="Maroon" title="c2">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp;  not <font color="Maroon" title="c3">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:66_1_1_3_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Maroon" title="c2">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> (  ex <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp;  not <font color="Maroon" title="c3">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:66_1_1_3_1"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Maroon" title="c3">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> <font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f2</font></span><br/>

<a NAME="E4:66_1_1_3_1"/>
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</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">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">w</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c4">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c5">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c4">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> , <font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c5">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_1_3_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">w</font>) = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">m</font></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="#E2:66_1_1_3_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">w</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_1_1_3_1"><span class="lab"><font color="Green" title="E37">A10</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>
<span class="kw">hence </span><a NAME="E5:66_1_1_3_1"/>
<font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T2" onmouseover="rs('binop_1/T2')" onmouseout="rh()">BINOP_1:2</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><span class="kw">assume </span><a NAME="E2:66_1_1_3"/>
( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  verum ) &amp; (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )</span><br/><span class="kw">hence </span><a NAME="E3:66_1_1_3"/>
( (  for <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  verum ) &amp; (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></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>
<span class="kw">hence </span><a NAME="E4:66_1_1"/>
( (  for <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  verum ) &amp; (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b3">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; (  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b4">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">v</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E2:66_1_1"><span class="lab"><font color="Green" title="E34">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>
