reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th47:
  ex S being 10-connectives non empty non void strict BoolSignature st
  S is 1-1-connectives (4,1) integer bool-correct & the carrier of S = {0,1} &
  ex I being SortSymbol of S st I = 1 &
  (the connectives of S).4 is_of_type {},I
  proof
    set X = {0,1}, Y = {0,1,2,3,4,5,6,7,8,9};
    reconsider 00 = 0, x1 = 1 as Element of X by TARSKI:def 2;
    reconsider y0 = 0, 01 = 1, 02 = 2, 03 = 3, 04 = 4, 05 = 5, 06 = 6,
    07 = 7, 08 = 8, 09 = 9 as Element of Y by ENUMSET1:def 8;
    set aa = <*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
    <*<*x1,x1*>*>;
    set a = aa+*({0}-->{});
    set r = ({0,1,2,9}-->00)\/({3,4,5,6,7,8}-->x1);
    <*00*> in X* & <*00,00*> in X* & <*x1*> in X* & <*x1,x1*> in X* &
    <*>X in X* by FINSEQ_1:def 11; then
A1: {{},<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>}
    c= X* &
    rng a c= rng aa \/ rng ({0}-->{}) & dom a =  dom aa \/ dom({0}-->{}) &
    dom ({0}-->{}) = {0} & rng ({0}-->{}) = {{}}
    by Th10,FUNCT_4:17,def 1; then
    rng a c= {<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>,
    <*x1,x1*>}\/{{}} by Th21; then
    rng a c= {{},<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>,
    <*x1,x1*>} & dom aa = Seg 9 by Th12,FINSEQ_1:89; then
    rng a c= X* & dom a = Y by A1,Th12,Th22; then
    reconsider a as Function of Y,X* by FUNCT_2:2;
A2: dom({0,1,2,9}-->00) = {0,1,2,9} & dom({3,4,5,6,7,8}-->x1) = {3,4,5,6,7,8};
    {0,1,2,9} misses {3,4,5,6,7,8}
    proof
      assume {0,1,2,9} meets {3,4,5,6,7,8}; then
      consider x being object such that
A3:   x in {0,1,2,9} & x in {3,4,5,6,7,8} by XBOOLE_0:3;
      (x = 0 or x = 1 or x = 2 or x = 9) & (x = 3 or x = 4 or x = 5 or x = 6 or
      x = 7 or x = 8) by A3,ENUMSET1:def 2,def 4;
      hence thesis;
    end; then
    reconsider r as Function by A2,GRFUNC_1:13;
A4: dom r = {0,1,2,9}\/{3,4,5,6,7,8} by A2,XTUPLE_0:23
    .= {0,1,2}\/{9}\/{3,4,5,6,7,8} by ENUMSET1:6
    .= {0,1,2}\/{3,4,5,6,7,8}\/{9} by XBOOLE_1:4
    .= {0,1,2,3,4,5,6,7,8}\/{9} by Th13
    .= Y by ENUMSET1:85;
    rng r = rng({0,1,2,9}-->00)\/rng({3,4,5,6,7,8}-->x1) by RELAT_1:12
    .={00}\/rng({3,4,5,6,7,8}-->x1) .= {00}\/{x1}
    .= X by ENUMSET1:1; then
    reconsider r as Function of Y,X by A4,FUNCT_2:2;
    set B = BoolSignature(#X,Y,a,r,00,<*y0,01,02,03,04,05,06,07*>^<*08,09*>#);
A5: len the connectives of B = len <*y0,01,02,03,04,05,06,07*>+len<*08,09*>
    by FINSEQ_1:22 .= 8+len <*08,09*> by Th19 .= 8+2 by FINSEQ_1:44 .= 10;
    B is 10-connectives non empty non void
    by A5; then
    reconsider B as 10-connectives non empty non void strict BoolSignature;
    take B;
    thus the connectives of B is one-to-one
    proof
      let x,y be object; assume
A6:   x in dom the connectives of B & y in dom the connectives of B;
      set c = the connectives of B;
A7:   dom c = Seg 10 by Th25;
A8:   y=1 or y=2 or y=3 or y=4 or y=5 or y=6 or y=7 or y=8 or y=9 or y=10
      by A6,A7,Th23,ENUMSET1:def 8;
      c.1 = y0 & c.2 = 01 & c.3 = 02 & c.4 = 03 & c.5 = 04 & c.6 = 05 &
      c.7 = 06 & c.8 = 07 & c.9 = 08 & c.10 = 09 by Th25;
      hence thesis by A7,A8,A6,Th23,ENUMSET1:def 8;
    end;
    thus B is (4,1) integer
    proof
      thus len the connectives of B >= 4+6 by A5;
      reconsider I = x1 as Element of B;
      take I; thus I = 1;
      thus I <> the bool-sort of B;
A9:   (the connectives of B).4 = 3 & 3 nin dom ({0}-->{})
      by Th25;
      hence (the Arity of B).((the connectives of B).4)
      = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
      <*<*x1,x1*>*>).3 by FUNCT_4:11 .= {} by Th24;
      3 in {3,4,5,6,7,8} & I in {I} by TARSKI:def 1,ENUMSET1:def 4; then
      [3,I] in {3,4,5,6,7,8}-->x1 by ZFMISC_1:87; then
      [3,I] in r by XBOOLE_0:def 3;
      hence (the ResultSort of B).((the connectives of B).4) = I
      by A9,FUNCT_1:1;
A10:   (the connectives of B).5 = 4 & 4 nin dom ({0}-->{})
      by Th25;
      hence (the Arity of B).((the connectives of B).(4+1))
      = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
      <*<*x1,x1*>*>).4 by FUNCT_4:11 .= {} by Th24;
      4 in {3,4,5,6,7,8} & I in {I} by TARSKI:def 1,ENUMSET1:def 4; then
      [4,I] in {3,4,5,6,7,8}-->x1 by ZFMISC_1:87; then
      [4,I] in r by XBOOLE_0:def 3;
      hence (the ResultSort of B).((the connectives of B).(4+1)) = I
      by A10,FUNCT_1:1;
      thus (the connectives of B).4 <> (the connectives of B).(4+1)
      by A9,Th25;
A11:   (the connectives of B).6 = 5 & 5 nin dom ({0}-->{})
      by Th25;
      hence (the Arity of B).((the connectives of B).(4+2))
      = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
      <*<*x1,x1*>*>).5 by FUNCT_4:11 .= <*I*> by Th24;
      5 in {3,4,5,6,7,8} & I in {I} by TARSKI:def 1,ENUMSET1:def 4; then
      [5,I] in {3,4,5,6,7,8}-->x1 by ZFMISC_1:87; then
      [5,I] in r by XBOOLE_0:def 3;
      hence (the ResultSort of B).((the connectives of B).(4+2)) = I
      by A11,FUNCT_1:1;
A12:   (the connectives of B).7 = 6 & 6 nin dom ({0}-->{})
      by Th25;
      hence (the Arity of B).((the connectives of B).(4+3))
      = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
      <*<*x1,x1*>*>).6 by FUNCT_4:11 .= <*I,I*> by Th24;
      6 in {3,4,5,6,7,8} & I in {I} by TARSKI:def 1,ENUMSET1:def 4; then
      [6,I] in {3,4,5,6,7,8}-->x1 by ZFMISC_1:87; then
      [6,I] in r by XBOOLE_0:def 3;
      hence (the ResultSort of B).((the connectives of B).(4+3)) = I
      by A12,FUNCT_1:1;
A13:  (the connectives of B).8 = 7 & 7 nin dom ({0}-->{})
      by Th25;
      hence (the Arity of B).((the connectives of B).(4+4))
      = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
      <*<*x1,x1*>*>).7 by FUNCT_4:11 .= <*I,I*> by Th24;
      7 in {3,4,5,6,7,8} & I in {I} by TARSKI:def 1,ENUMSET1:def 4; then
      [7,I] in {3,4,5,6,7,8}-->x1 by ZFMISC_1:87; then
      [7,I] in r by XBOOLE_0:def 3;
      hence (the ResultSort of B).((the connectives of B).(4+4)) = I
      by A13,FUNCT_1:1;
A14:   (the connectives of B).9 = 8 & 8 nin dom ({0}-->{})
      by Th25;
      hence (the Arity of B).((the connectives of B).(4+5))
      = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
      <*<*x1,x1*>*>).8 by FUNCT_4:11 .= <*I,I*> by Th24;
      8 in {3,4,5,6,7,8} & I in {I} by TARSKI:def 1,ENUMSET1:def 4; then
      [8,I] in {3,4,5,6,7,8}-->x1 by ZFMISC_1:87; then
      [8,I] in r by XBOOLE_0:def 3;
      hence (the ResultSort of B).((the connectives of B).(4+5)) = I
      by A14,FUNCT_1:1;
      thus (the connectives of B).(4+3) <> (the connectives of B).(4+4)
      by A13,Th25;
      thus (the connectives of B).(4+3) <> (the connectives of B).(4+5)
      by A14,Th25;
      thus (the connectives of B).(4+4) <> (the connectives of B).(4+5)
      by A14,Th25;
A15:   (the connectives of B).10 = 9 & 9 nin dom ({0}-->{})
      by Th25;
      hence (the Arity of B).((the connectives of B).(4+6))
      = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
      <*<*x1,x1*>*>).9 by FUNCT_4:11 .= <*I,I*> by Th24;
      9 in {0,1,2,9} & the bool-sort of B in {the bool-sort of B}
      by TARSKI:def 1,ENUMSET1:def 2; then
      [9,00] in {0,1,2,9}-->00 by ZFMISC_1:87; then
      [9,00] in r by XBOOLE_0:def 3;
      hence (the ResultSort of B).((the connectives of B).(4+6)) =
      the bool-sort of B by A15,FUNCT_1:1;
    end;
    thus len the connectives of B >= 3 by A5;
A16: (the connectives of B).1 = 0 & 0 in {0} by Th25,TARSKI:def 1;
    hence (the Arity of B).((the connectives of B).1)
    = ({0}-->{}).0 by A1,FUNCT_4:13
    .= {};
    0 in {0,1,2,9} & 00 in {00} by TARSKI:def 1,ENUMSET1:def 2; then
    [0,the bool-sort of B] in {0,1,2,9}-->00 by ZFMISC_1:87; then
    [0,the bool-sort of B] in r by XBOOLE_0:def 3;
    hence (the ResultSort of B).((the connectives of B).1) = the bool-sort of B
    by A16,FUNCT_1:1;
A17: (the connectives of B).2 = 1 & 1 nin {0} by Th25,TARSKI:def 1;
    hence (the Arity of B).((the connectives of B).2)
    = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
    <*<*x1,x1*>*>).1 by A1,FUNCT_4:11 .= <*the bool-sort of B*> by Th24;
    1 in {0,1,2,9} & 00 in {00} by TARSKI:def 1,ENUMSET1:def 2; then
    [1,the bool-sort of B] in {0,1,2,9}-->00 by ZFMISC_1:87; then
    [1,the bool-sort of B] in r by XBOOLE_0:def 3;
    hence (the ResultSort of B).((the connectives of B).2) = the bool-sort of B
    by A17,FUNCT_1:1;
A18: (the connectives of B).3 = 2 & 2 nin {0} by Th25,TARSKI:def 1;
    hence (the Arity of B).((the connectives of B).3)
    = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
    <*<*x1,x1*>*>).2 by A1,FUNCT_4:11
    .= <*the bool-sort of B,the bool-sort of B*> by Th24;
    2 in {0,1,2,9} & 00 in {00} by TARSKI:def 1,ENUMSET1:def 2; then
    [2,the bool-sort of B] in {0,1,2,9}-->00 by ZFMISC_1:87; then
    [2,the bool-sort of B] in r by XBOOLE_0:def 3;
    hence (the ResultSort of B).((the connectives of B).3) = the bool-sort of B
    by A18,FUNCT_1:1;
    thus the carrier of B = {0,1};
    reconsider I = 1 as SortSymbol of B;
    take I; thus I = 1;
A19: (the connectives of B).4 = 3 & 3 nin dom ({0}-->{})
    by Th25;
    hence (the Arity of B).((the connectives of B).4)
    = (<*<*00*>,<*00,00*>,{},{},<*x1*>,<*x1,x1*>,<*x1,x1*>,<*x1,x1*>*>^
    <*<*x1,x1*>*>).3 by FUNCT_4:11 .= {} by Th24;
    3 in {3,4,5,6,7,8} & I in {I} by TARSKI:def 1,ENUMSET1:def 4; then
    [3,I] in {3,4,5,6,7,8}-->x1 by ZFMISC_1:87; then
    [3,I] in r by XBOOLE_0:def 3;
    hence (the ResultSort of B).((the connectives of B).4) = I
    by A19,FUNCT_1:1;
  end;
