bvfunc_6.miz

    begin

    reserve Y for non empty set;

    theorem :: BVFUNC_6:1

    for a,b be Function of Y, BOOLEAN holds (a 'imp' (b 'imp' (a '&' b))) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((a 'imp' (b 'imp' (a '&' b))) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' (b 'imp' (a '&' b))) . x) = (( 'not' (a . x)) 'or' ((b 'imp' (a '&' b)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' ((a '&' b) . x))) by BVFUNC_1:def 8

        .= (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' ((a . x) '&' (b . x)))) by MARGREL1:def 20

        .= (( 'not' (a . x)) 'or' ((( 'not' (b . x)) 'or' (a . x)) '&' (( 'not' (b . x)) 'or' (b . x)))) by XBOOLEAN: 9

        .= (( 'not' (a . x)) 'or' ( TRUE '&' (( 'not' (b . x)) 'or' (a . x)))) by XBOOLEAN: 102

        .= ((( 'not' (a . x)) 'or' (a . x)) 'or' ( 'not' (b . x)))

        .= ( TRUE 'or' ( 'not' (b . x))) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:2

    for a,b be Function of Y, BOOLEAN holds ((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))) . x) = (( 'not' ((a 'imp' b) . x)) 'or' (((b 'imp' a) 'imp' (a 'eqv' b)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (b . x))) 'or' (((b 'imp' a) 'imp' (a 'eqv' b)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (b . x))) 'or' (( 'not' ((b 'imp' a) . x)) 'or' ((a 'eqv' b) . x))) by BVFUNC_1:def 8

        .= ((( 'not' ( 'not' (a . x))) '&' ( 'not' (b . x))) 'or' (( 'not' (( 'not' (b . x)) 'or' (a . x))) 'or' ((a 'eqv' b) . x))) by BVFUNC_1:def 8

        .= (((a . x) '&' ( 'not' (b . x))) 'or' (((b . x) '&' ( 'not' (a . x))) 'or' ( 'not' ((a . x) 'xor' (b . x))))) by BVFUNC_1:def 9

        .= (((a . x) '&' ( 'not' (b . x))) 'or' (((( 'not' (a . x)) '&' (b . x)) 'or' ( 'not' (( 'not' (a . x)) '&' (b . x)))) '&' ((( 'not' (a . x)) '&' (b . x)) 'or' ( 'not' ((a . x) '&' ( 'not' (b . x))))))) by XBOOLEAN: 9

        .= (((a . x) '&' ( 'not' (b . x))) 'or' ( TRUE '&' ((( 'not' (a . x)) '&' (b . x)) 'or' ( 'not' ((a . x) '&' ( 'not' (b . x))))))) by XBOOLEAN: 102

        .= ((((a . x) '&' ( 'not' (b . x))) 'or' ( 'not' ((a . x) '&' ( 'not' (b . x))))) 'or' (( 'not' (a . x)) '&' (b . x)))

        .= ( TRUE 'or' (( 'not' (a . x)) '&' (b . x))) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:3

    for a,b be Function of Y, BOOLEAN holds ((a 'or' b) 'eqv' (b 'or' a)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'or' b) 'eqv' (b 'or' a)) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'or' b) 'eqv' (b 'or' a)) . x) = ( 'not' (((a 'or' b) . x) 'xor' ((b 'or' a) . x))) by BVFUNC_1:def 9

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:4

    for a,b,c be Function of Y, BOOLEAN holds (((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))) . x) = (( 'not' (((a '&' b) 'imp' c) . x)) 'or' ((a 'imp' (b 'imp' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' ((a '&' b) . x)) 'or' (c . x))) 'or' ((a 'imp' (b 'imp' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' ((a . x) '&' (b . x))) 'or' (c . x))) 'or' ((a 'imp' (b 'imp' c)) . x)) by MARGREL1:def 20

        .= (( 'not' (( 'not' ((a . x) '&' (b . x))) 'or' (c . x))) 'or' (( 'not' (a . x)) 'or' ((b 'imp' c) . x))) by BVFUNC_1:def 8

        .= (( 'not' ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (c . x))) 'or' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (c . x)))) by BVFUNC_1:def 8

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:5

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)) . x) = (( 'not' ((a 'imp' (b 'imp' c)) . x)) 'or' (((a '&' b) 'imp' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' ((b 'imp' c) . x))) 'or' (((a '&' b) 'imp' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (c . x)))) 'or' (((a '&' b) 'imp' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (c . x)))) 'or' (( 'not' ((a '&' b) . x)) 'or' (c . x))) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (c . x)))) 'or' ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (c . x))) by MARGREL1:def 20

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:6

    for a,b,c be Function of Y, BOOLEAN holds ((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b)))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b)))) . x) = TRUE

      proof

        let x be Element of Y;

        (((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b)))) . x) = (( 'not' ((c 'imp' a) . x)) 'or' (((c 'imp' b) 'imp' (c 'imp' (a '&' b))) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (c . x)) 'or' (a . x))) 'or' (((c 'imp' b) 'imp' (c 'imp' (a '&' b))) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (c . x)) 'or' (a . x))) 'or' (( 'not' ((c 'imp' b) . x)) 'or' ((c 'imp' (a '&' b)) . x))) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (c . x)) 'or' (a . x))) 'or' (( 'not' (( 'not' (c . x)) 'or' (b . x))) 'or' ((c 'imp' (a '&' b)) . x))) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (c . x)) 'or' (a . x))) 'or' (( 'not' (( 'not' (c . x)) 'or' (b . x))) 'or' (( 'not' (c . x)) 'or' ((a '&' b) . x)))) by BVFUNC_1:def 8

        .= (((c . x) '&' ( 'not' (a . x))) 'or' ((( 'not' ( 'not' (c . x))) '&' ( 'not' (b . x))) 'or' (( 'not' (c . x)) 'or' ((a . x) '&' (b . x))))) by MARGREL1:def 20

        .= (((c . x) '&' ( 'not' (a . x))) 'or' (((c . x) '&' ( 'not' (b . x))) 'or' ((( 'not' (c . x)) 'or' (a . x)) '&' (( 'not' (c . x)) 'or' (b . x))))) by XBOOLEAN: 9

        .= (((c . x) '&' ( 'not' (a . x))) 'or' ((((c . x) '&' ( 'not' (b . x))) 'or' (( 'not' (c . x)) 'or' (a . x))) '&' (((c . x) '&' ( 'not' (b . x))) 'or' (( 'not' (c . x)) 'or' ( 'not' ( 'not' (b . x))))))) by XBOOLEAN: 9

        .= (((c . x) '&' ( 'not' (a . x))) 'or' ( TRUE '&' (((c . x) '&' ( 'not' (b . x))) 'or' (( 'not' (c . x)) 'or' (a . x))))) by XBOOLEAN: 102

        .= ((((c . x) '&' ( 'not' (a . x))) 'or' ( 'not' ((c . x) '&' ( 'not' (a . x))))) 'or' ((c . x) '&' ( 'not' (b . x))))

        .= ( TRUE 'or' ((c . x) '&' ( 'not' (b . x)))) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:7

    for a,b,c be Function of Y, BOOLEAN holds (((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))) . x) = (( 'not' (((a 'or' b) 'imp' c) . x)) 'or' (((a 'imp' c) 'or' (b 'imp' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' ((a 'or' b) . x)) 'or' (c . x))) 'or' (((a 'imp' c) 'or' (b 'imp' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' ((a . x) 'or' (b . x))) 'or' (c . x))) 'or' (((a 'imp' c) 'or' (b 'imp' c)) . x)) by BVFUNC_1:def 4

        .= (( 'not' (( 'not' ((a . x) 'or' (b . x))) 'or' (c . x))) 'or' (((a 'imp' c) . x) 'or' ((b 'imp' c) . x))) by BVFUNC_1:def 4

        .= (( 'not' (( 'not' ((a . x) 'or' (b . x))) 'or' (c . x))) 'or' ((( 'not' (a . x)) 'or' (c . x)) 'or' ((b 'imp' c) . x))) by BVFUNC_1:def 8

        .= ((( 'not' ( 'not' ((a . x) 'or' (b . x)))) '&' ( 'not' (c . x))) 'or' ((( 'not' (a . x)) 'or' (c . x)) 'or' (( 'not' (b . x)) 'or' (c . x)))) by BVFUNC_1:def 8

        .= ((((b . x) '&' ( 'not' (c . x))) 'or' ((a . x) '&' ( 'not' (c . x)))) 'or' (( 'not' ((a . x) '&' ( 'not' (c . x)))) 'or' (( 'not' (b . x)) 'or' (c . x)))) by XBOOLEAN: 8

        .= ((((b . x) '&' ( 'not' (c . x))) 'or' (((a . x) '&' ( 'not' (c . x))) 'or' ( 'not' ((a . x) '&' ( 'not' (c . x)))))) 'or' (( 'not' (b . x)) 'or' (c . x)))

        .= ((((b . x) '&' ( 'not' (c . x))) 'or' TRUE ) 'or' (( 'not' (b . x)) 'or' (c . x))) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:8

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))) . x) = (( 'not' ((a 'imp' c) . x)) 'or' (((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' ((b 'imp' c) . x)) 'or' (((a 'or' b) 'imp' c) . x))) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (((a 'or' b) 'imp' c) . x))) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (( 'not' ((a 'or' b) . x)) 'or' (c . x)))) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (( 'not' ((a . x) 'or' (b . x))) 'or' (c . x)))) by BVFUNC_1:def 4

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (((c . x) 'or' ( 'not' (a . x))) '&' (( 'not' (b . x)) 'or' (c . x))))) by XBOOLEAN: 9

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' ((( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' ((c . x) 'or' ( 'not' (a . x)))) '&' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (( 'not' (b . x)) 'or' (c . x))))) by XBOOLEAN: 9

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' ( TRUE '&' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' ((c . x) 'or' ( 'not' (a . x)))))) by XBOOLEAN: 102

        .= (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (( 'not' (a . x)) 'or' (c . x))))

        .= ((( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (a . x)) 'or' (c . x))) 'or' ( 'not' (( 'not' (b . x)) 'or' (c . x))))

        .= ( TRUE 'or' ( 'not' (( 'not' (b . x)) 'or' (c . x)))) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:9



     Th9: for a,b,c be Function of Y, BOOLEAN holds (((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)) . x) = (( 'not' (((a 'imp' c) '&' (b 'imp' c)) . x)) 'or' (((a 'or' b) 'imp' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' (((a 'imp' c) . x) '&' ((b 'imp' c) . x))) 'or' (((a 'or' b) 'imp' c) . x)) by MARGREL1:def 20

        .= (( 'not' ((( 'not' (a . x)) 'or' (c . x)) '&' ((b 'imp' c) . x))) 'or' (((a 'or' b) 'imp' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((( 'not' (a . x)) 'or' (c . x)) '&' (( 'not' (b . x)) 'or' (c . x)))) 'or' (((a 'or' b) 'imp' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((( 'not' (a . x)) 'or' (c . x)) '&' (( 'not' (b . x)) 'or' (c . x)))) 'or' (( 'not' ((a 'or' b) . x)) 'or' (c . x))) by BVFUNC_1:def 8

        .= ((( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' ( 'not' (( 'not' (b . x)) 'or' (c . x)))) 'or' ((c . x) 'or' ( 'not' ((a . x) 'or' (b . x))))) by BVFUNC_1:def 4

        .= ((( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' ( 'not' (( 'not' (b . x)) 'or' (c . x)))) 'or' ((( 'not' (a . x)) 'or' (c . x)) '&' ((c . x) 'or' ( 'not' (b . x))))) by XBOOLEAN: 9

        .= (((( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' ( 'not' (( 'not' (b . x)) 'or' (c . x)))) 'or' (( 'not' (a . x)) 'or' (c . x))) '&' ((( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' ( 'not' (( 'not' (b . x)) 'or' (c . x)))) 'or' (( 'not' (b . x)) 'or' (c . x)))) by XBOOLEAN: 9

        .= ((( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (a . x)) 'or' (c . x)))) '&' (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (( 'not' (b . x)) 'or' (c . x)))))

        .= ((( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' TRUE ) '&' (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (( 'not' (b . x)) 'or' (c . x))) 'or' (( 'not' (b . x)) 'or' (c . x))))) by XBOOLEAN: 102

        .= ( TRUE '&' (( 'not' (( 'not' (a . x)) 'or' (c . x))) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:10

    for a,b be Function of Y, BOOLEAN holds ((a 'imp' (b '&' ( 'not' b))) 'imp' ( 'not' a)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'imp' (b '&' ( 'not' b))) 'imp' ( 'not' a)) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'imp' (b '&' ( 'not' b))) 'imp' ( 'not' a)) . x) = (( 'not' ((a 'imp' (b '&' ( 'not' b))) . x)) 'or' (( 'not' a) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' ((b '&' ( 'not' b)) . x))) 'or' (( 'not' a) . x)) by BVFUNC_1:def 8

        .= (((a . x) '&' (( 'not' (b . x)) 'or' ( 'not' (( 'not' b) . x)))) 'or' (( 'not' a) . x)) by MARGREL1:def 20

        .= (((a . x) '&' (( 'not' (b . x)) 'or' ( 'not' ( 'not' (b . x))))) 'or' (( 'not' a) . x)) by MARGREL1:def 19

        .= (((a . x) '&' TRUE ) 'or' (( 'not' a) . x)) by XBOOLEAN: 102

        .= ((a . x) 'or' ( 'not' (a . x))) by MARGREL1:def 19

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:11

    for a,b,c be Function of Y, BOOLEAN holds (((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))) . x) = (( 'not' (((a 'or' b) '&' (a 'or' c)) . x)) 'or' ((a 'or' (b '&' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (((a 'or' b) . x) '&' ((a 'or' c) . x))) 'or' ((a 'or' (b '&' c)) . x)) by MARGREL1:def 20

        .= (( 'not' (((a . x) 'or' (b . x)) '&' ((a 'or' c) . x))) 'or' ((a 'or' (b '&' c)) . x)) by BVFUNC_1:def 4

        .= (( 'not' (((a . x) 'or' (b . x)) '&' ((a . x) 'or' (c . x)))) 'or' ((a 'or' (b '&' c)) . x)) by BVFUNC_1:def 4

        .= (( 'not' (((a . x) 'or' (b . x)) '&' ((a . x) 'or' (c . x)))) 'or' ((a . x) 'or' ((b '&' c) . x))) by BVFUNC_1:def 4

        .= (( 'not' (((a . x) 'or' (b . x)) '&' ((a . x) 'or' (c . x)))) 'or' ((a . x) 'or' ((b . x) '&' (c . x)))) by MARGREL1:def 20

        .= ((( 'not' ((a . x) 'or' (b . x))) 'or' ( 'not' ((a . x) 'or' (c . x)))) 'or' (((a . x) 'or' (b . x)) '&' ((a . x) 'or' (c . x)))) by XBOOLEAN: 9

        .= (((( 'not' ((a . x) 'or' (c . x))) 'or' ( 'not' ((a . x) 'or' (b . x)))) 'or' ((a . x) 'or' (b . x))) '&' ((( 'not' ((a . x) 'or' (b . x))) 'or' ( 'not' ((a . x) 'or' (c . x)))) 'or' ((a . x) 'or' (c . x)))) by XBOOLEAN: 9

        .= ((( 'not' ((a . x) 'or' (c . x))) 'or' (( 'not' ((a . x) 'or' (b . x))) 'or' ((a . x) 'or' (b . x)))) '&' (( 'not' ((a . x) 'or' (b . x))) 'or' (( 'not' ((a . x) 'or' (c . x))) 'or' ((a . x) 'or' (c . x)))))

        .= ((( 'not' ((a . x) 'or' (c . x))) 'or' TRUE ) '&' (( 'not' ((a . x) 'or' (b . x))) 'or' (( 'not' ((a . x) 'or' (c . x))) 'or' ((a . x) 'or' (c . x))))) by XBOOLEAN: 102

        .= ( TRUE '&' (( 'not' ((a . x) 'or' (b . x))) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:12

    for a,b,c be Function of Y, BOOLEAN holds ((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))) . x) = TRUE

      proof

        let x be Element of Y;

        (((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))) . x) = (( 'not' ((a '&' (b 'or' c)) . x)) 'or' (((a '&' b) 'or' (a '&' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) '&' ((b 'or' c) . x))) 'or' (((a '&' b) 'or' (a '&' c)) . x)) by MARGREL1:def 20

        .= (( 'not' ((a . x) '&' ((b . x) 'or' (c . x)))) 'or' (((a '&' b) 'or' (a '&' c)) . x)) by BVFUNC_1:def 4

        .= (( 'not' ((a . x) '&' ((b . x) 'or' (c . x)))) 'or' (((a '&' b) . x) 'or' ((a '&' c) . x))) by BVFUNC_1:def 4

        .= (( 'not' ((a . x) '&' ((b . x) 'or' (c . x)))) 'or' (((a . x) '&' (b . x)) 'or' ((a '&' c) . x))) by MARGREL1:def 20

        .= (( 'not' ((a . x) '&' ((b . x) 'or' (c . x)))) 'or' (((a . x) '&' (b . x)) 'or' ((a . x) '&' (c . x)))) by MARGREL1:def 20

        .= ((((a . x) '&' (b . x)) 'or' ((a . x) '&' (c . x))) 'or' (( 'not' ((a . x) '&' (b . x))) '&' ( 'not' ((a . x) '&' (c . x))))) by XBOOLEAN: 8

        .= (((((a . x) '&' (c . x)) 'or' ((a . x) '&' (b . x))) 'or' ( 'not' ((a . x) '&' (b . x)))) '&' ((((a . x) '&' (b . x)) 'or' ((a . x) '&' (c . x))) 'or' ( 'not' ((a . x) '&' (c . x))))) by XBOOLEAN: 9

        .= ((((a . x) '&' (c . x)) 'or' (((a . x) '&' (b . x)) 'or' ( 'not' ((a . x) '&' (b . x))))) '&' (((a . x) '&' (b . x)) 'or' (((a . x) '&' (c . x)) 'or' ( 'not' ((a . x) '&' (c . x))))))

        .= ((((a . x) '&' (c . x)) 'or' TRUE ) '&' (((a . x) '&' (b . x)) 'or' (((a . x) '&' (c . x)) 'or' ( 'not' ((a . x) '&' (c . x)))))) by XBOOLEAN: 102

        .= ( TRUE '&' (((a . x) '&' (b . x)) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:13

    for a,b,c be Function of Y, BOOLEAN holds (((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)) . x) = (( 'not' (((a 'or' c) '&' (b 'or' c)) . x)) 'or' (((a '&' b) 'or' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' (((a 'or' c) . x) '&' ((b 'or' c) . x))) 'or' (((a '&' b) 'or' c) . x)) by MARGREL1:def 20

        .= (( 'not' (((a . x) 'or' (c . x)) '&' ((b 'or' c) . x))) 'or' (((a '&' b) 'or' c) . x)) by BVFUNC_1:def 4

        .= ((( 'not' ((a . x) 'or' (c . x))) 'or' ( 'not' ((b . x) 'or' (c . x)))) 'or' (((a '&' b) 'or' c) . x)) by BVFUNC_1:def 4

        .= ((( 'not' ((a . x) 'or' (c . x))) 'or' ( 'not' ((b . x) 'or' (c . x)))) 'or' (((a '&' b) . x) 'or' (c . x))) by BVFUNC_1:def 4

        .= ((( 'not' ((a . x) 'or' (c . x))) 'or' ( 'not' ((b . x) 'or' (c . x)))) 'or' ((c . x) 'or' ((a . x) '&' (b . x)))) by MARGREL1:def 20

        .= ((( 'not' ((a . x) 'or' (c . x))) 'or' ( 'not' ((b . x) 'or' (c . x)))) 'or' (((a . x) 'or' (c . x)) '&' ((c . x) 'or' (b . x)))) by XBOOLEAN: 9

        .= (((( 'not' ((a . x) 'or' (c . x))) 'or' ( 'not' ((b . x) 'or' (c . x)))) 'or' ((a . x) 'or' (c . x))) '&' ((( 'not' ((a . x) 'or' (c . x))) 'or' ( 'not' ((b . x) 'or' (c . x)))) 'or' ((b . x) 'or' (c . x)))) by XBOOLEAN: 9

        .= ((( 'not' ((b . x) 'or' (c . x))) 'or' (( 'not' ((a . x) 'or' (c . x))) 'or' ((a . x) 'or' (c . x)))) '&' (( 'not' ((a . x) 'or' (c . x))) 'or' (( 'not' ((b . x) 'or' (c . x))) 'or' ((b . x) 'or' (c . x)))))

        .= ((( 'not' ((b . x) 'or' (c . x))) 'or' TRUE ) '&' (( 'not' ((a . x) 'or' (c . x))) 'or' (( 'not' ((b . x) 'or' (c . x))) 'or' ((b . x) 'or' (c . x))))) by XBOOLEAN: 102

        .= ( TRUE '&' (( 'not' ((a . x) 'or' (c . x))) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:14

    for a,b,c be Function of Y, BOOLEAN holds (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))) . x) = (( 'not' (((a 'or' b) '&' c) . x)) 'or' (((a '&' c) 'or' (b '&' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (((a 'or' b) . x) '&' (c . x))) 'or' (((a '&' c) 'or' (b '&' c)) . x)) by MARGREL1:def 20

        .= (( 'not' (((a . x) 'or' (b . x)) '&' (c . x))) 'or' (((a '&' c) 'or' (b '&' c)) . x)) by BVFUNC_1:def 4

        .= (( 'not' (((a . x) 'or' (b . x)) '&' (c . x))) 'or' (((a '&' c) . x) 'or' ((b '&' c) . x))) by BVFUNC_1:def 4

        .= (( 'not' (((a . x) 'or' (b . x)) '&' (c . x))) 'or' (((a . x) '&' (c . x)) 'or' ((b '&' c) . x))) by MARGREL1:def 20

        .= (( 'not' ((c . x) '&' ((a . x) 'or' (b . x)))) 'or' (((a . x) '&' (c . x)) 'or' ((b . x) '&' (c . x)))) by MARGREL1:def 20

        .= ((((a . x) '&' (c . x)) 'or' ((b . x) '&' (c . x))) 'or' (( 'not' ((a . x) '&' (c . x))) '&' ( 'not' ((b . x) '&' (c . x))))) by XBOOLEAN: 8

        .= (((((b . x) '&' (c . x)) 'or' ((a . x) '&' (c . x))) 'or' ( 'not' ((a . x) '&' (c . x)))) '&' ((((a . x) '&' (c . x)) 'or' ((b . x) '&' (c . x))) 'or' ( 'not' ((b . x) '&' (c . x))))) by XBOOLEAN: 9

        .= ((((b . x) '&' (c . x)) 'or' (((a . x) '&' (c . x)) 'or' ( 'not' ((a . x) '&' (c . x))))) '&' (((a . x) '&' (c . x)) 'or' (((b . x) '&' (c . x)) 'or' ( 'not' ((b . x) '&' (c . x))))))

        .= ((((b . x) '&' (c . x)) 'or' TRUE ) '&' (((a . x) '&' (c . x)) 'or' (((b . x) '&' (c . x)) 'or' ( 'not' ((b . x) '&' (c . x)))))) by XBOOLEAN: 102

        .= ( TRUE '&' (((a . x) '&' (c . x)) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:15

    for a,b be Function of Y, BOOLEAN holds (a '&' b) = ( I_el Y) implies (a 'or' b) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      assume

       A1: (a '&' b) = ( I_el Y);

      for x be Element of Y holds ((a 'or' b) . x) = TRUE

      proof

        let x be Element of Y;

        ((a '&' b) . x) = TRUE by A1, BVFUNC_1:def 11;

        then

         A2: ((a . x) '&' (b . x)) = TRUE by MARGREL1:def 20;

        then (a . x) = TRUE by MARGREL1: 12;



        then ((a 'or' b) . x) = ( TRUE 'or' TRUE ) by A2, BVFUNC_1:def 4

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:16

    for a,b,c be Function of Y, BOOLEAN holds (a 'imp' b) = ( I_el Y) implies ((a 'or' c) 'imp' (b 'or' c)) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      assume

       A1: (a 'imp' b) = ( I_el Y);

      for x be Element of Y holds (((a 'or' c) 'imp' (b 'or' c)) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' b) . x) = TRUE by A1, BVFUNC_1:def 11;

        then

         A2: (( 'not' (a . x)) 'or' (b . x)) = TRUE by BVFUNC_1:def 8;

        (((a 'or' c) 'imp' (b 'or' c)) . x) = (( 'not' ((a 'or' c) . x)) 'or' ((b 'or' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) 'or' (c . x))) 'or' ((b 'or' c) . x)) by BVFUNC_1:def 4

        .= (((b . x) 'or' (c . x)) 'or' (( 'not' (a . x)) '&' ( 'not' (c . x)))) by BVFUNC_1:def 4

        .= ((((c . x) 'or' (b . x)) 'or' ( 'not' (a . x))) '&' (((b . x) 'or' (c . x)) 'or' ( 'not' (c . x)))) by XBOOLEAN: 9

        .= (((c . x) 'or' (( 'not' (a . x)) 'or' (b . x))) '&' (((b . x) 'or' (c . x)) 'or' ( 'not' (c . x))))

        .= ( TRUE '&' ((b . x) 'or' ((c . x) 'or' ( 'not' (c . x))))) by A2

        .= ( TRUE '&' ((b . x) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:17

    for a,b,c be Function of Y, BOOLEAN holds (a 'imp' b) = ( I_el Y) implies ((a '&' c) 'imp' (b '&' c)) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      assume

       A1: (a 'imp' b) = ( I_el Y);

      for x be Element of Y holds (((a '&' c) 'imp' (b '&' c)) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' b) . x) = TRUE by A1, BVFUNC_1:def 11;

        then

         A2: (( 'not' (a . x)) 'or' (b . x)) = TRUE by BVFUNC_1:def 8;

        (((a '&' c) 'imp' (b '&' c)) . x) = (( 'not' ((a '&' c) . x)) 'or' ((b '&' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) '&' (c . x))) 'or' ((b '&' c) . x)) by MARGREL1:def 20

        .= ((( 'not' (a . x)) 'or' ( 'not' (c . x))) 'or' ((b . x) '&' (c . x))) by MARGREL1:def 20

        .= (((( 'not' (c . x)) 'or' ( 'not' (a . x))) 'or' (b . x)) '&' ((( 'not' (a . x)) 'or' ( 'not' (c . x))) 'or' (c . x))) by XBOOLEAN: 9

        .= ((( 'not' (c . x)) 'or' (( 'not' (a . x)) 'or' (b . x))) '&' (( 'not' (a . x)) 'or' (( 'not' (c . x)) 'or' (c . x))))

        .= ((( 'not' (c . x)) 'or' (( 'not' (a . x)) 'or' (b . x))) '&' (( 'not' (a . x)) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE by A2;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:18



     th18: for a,b,c be Function of Y, BOOLEAN holds (c 'imp' a) = ( I_el Y) & (c 'imp' b) = ( I_el Y) implies (c 'imp' (a '&' b)) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      assume that

       A1: (c 'imp' a) = ( I_el Y) and

       A2: (c 'imp' b) = ( I_el Y);

      for x be Element of Y holds ((c 'imp' (a '&' b)) . x) = TRUE

      proof

        let x be Element of Y;

        ((c 'imp' a) . x) = TRUE by A1, BVFUNC_1:def 11;

        then

         A3: (( 'not' (c . x)) 'or' (a . x)) = TRUE by BVFUNC_1:def 8;

        ((c 'imp' b) . x) = TRUE by A2, BVFUNC_1:def 11;

        then

         A4: (( 'not' (c . x)) 'or' (b . x)) = TRUE by BVFUNC_1:def 8;

        ((c 'imp' (a '&' b)) . x) = (( 'not' (c . x)) 'or' ((a '&' b) . x)) by BVFUNC_1:def 8

        .= (( 'not' (c . x)) 'or' ((a . x) '&' (b . x))) by MARGREL1:def 20

        .= ( TRUE '&' TRUE ) by A3, A4, XBOOLEAN: 9

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:19

    for a,b,c be Function of Y, BOOLEAN holds (a 'imp' c) = ( I_el Y) & (b 'imp' c) = ( I_el Y) implies ((a 'or' b) 'imp' c) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      assume that

       A1: (a 'imp' c) = ( I_el Y) and

       A2: (b 'imp' c) = ( I_el Y);

      for x be Element of Y holds (((a 'or' b) 'imp' c) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' c) . x) = TRUE by A1, BVFUNC_1:def 11;

        then

         A3: (( 'not' (a . x)) 'or' (c . x)) = TRUE by BVFUNC_1:def 8;

        ((b 'imp' c) . x) = TRUE by A2, BVFUNC_1:def 11;

        then

         A4: (( 'not' (b . x)) 'or' (c . x)) = TRUE by BVFUNC_1:def 8;

        (((a 'or' b) 'imp' c) . x) = (( 'not' ((a 'or' b) . x)) 'or' (c . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) 'or' (b . x))) 'or' (c . x)) by BVFUNC_1:def 4

        .= ( TRUE '&' TRUE ) by A3, A4, XBOOLEAN: 9

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:20

    for a,b be Function of Y, BOOLEAN holds (a 'or' b) = ( I_el Y) & ( 'not' a) = ( I_el Y) implies b = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      assume that

       A1: (a 'or' b) = ( I_el Y) and

       A2: ( 'not' a) = ( I_el Y);

      for x be Element of Y holds (b . x) = TRUE

      proof

        let x be Element of Y;

        (( 'not' a) . x) = TRUE by A2, BVFUNC_1:def 11;

        then

         A3: ( 'not' (a . x)) = TRUE by MARGREL1:def 19;

        ((a 'or' b) . x) = TRUE by A1, BVFUNC_1:def 11;

        then ((a . x) 'or' (b . x)) = TRUE by BVFUNC_1:def 4;

        hence thesis by A3;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:21



     tt: for a,b,c,d be Function of Y, BOOLEAN holds (a 'imp' b) = ( I_el Y) & (c 'imp' d) = ( I_el Y) implies ((a '&' c) 'imp' (b '&' d)) = ( I_el Y)

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      assume that

       A1: (a 'imp' b) = ( I_el Y) and

       A2: (c 'imp' d) = ( I_el Y);

      for x be Element of Y holds (((a '&' c) 'imp' (b '&' d)) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' b) . x) = TRUE by A1, BVFUNC_1:def 11;

        then

         A3: (( 'not' (a . x)) 'or' (b . x)) = TRUE by BVFUNC_1:def 8;

        ((c 'imp' d) . x) = TRUE by A2, BVFUNC_1:def 11;

        then

         A4: (( 'not' (c . x)) 'or' (d . x)) = TRUE by BVFUNC_1:def 8;

        (((a '&' c) 'imp' (b '&' d)) . x) = (( 'not' ((a '&' c) . x)) 'or' ((b '&' d) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) '&' (c . x))) 'or' ((b '&' d) . x)) by MARGREL1:def 20

        .= ((( 'not' (a . x)) 'or' ( 'not' (c . x))) 'or' ((b . x) '&' (d . x))) by MARGREL1:def 20

        .= (((( 'not' (c . x)) 'or' ( 'not' (a . x))) 'or' (b . x)) '&' ((( 'not' (a . x)) 'or' ( 'not' (c . x))) 'or' (d . x))) by XBOOLEAN: 9

        .= ((( 'not' (c . x)) 'or' (( 'not' (a . x)) 'or' (b . x))) '&' ((( 'not' (a . x)) 'or' ( 'not' (c . x))) 'or' (d . x)))

        .= ( TRUE '&' (( 'not' (a . x)) 'or' TRUE )) by A3, A4, BINARITH: 11

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:22



     Th22: for a,b,c,d be Function of Y, BOOLEAN holds (a 'imp' b) = ( I_el Y) & (c 'imp' d) = ( I_el Y) implies ((a 'or' c) 'imp' (b 'or' d)) = ( I_el Y)

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      assume that

       A1: (a 'imp' b) = ( I_el Y) and

       A2: (c 'imp' d) = ( I_el Y);

      for x be Element of Y holds (((a 'or' c) 'imp' (b 'or' d)) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' b) . x) = TRUE by A1, BVFUNC_1:def 11;

        then

         A3: (( 'not' (a . x)) 'or' (b . x)) = TRUE by BVFUNC_1:def 8;

        ((c 'imp' d) . x) = TRUE by A2, BVFUNC_1:def 11;

        then

         A4: (( 'not' (c . x)) 'or' (d . x)) = TRUE by BVFUNC_1:def 8;

        (((a 'or' c) 'imp' (b 'or' d)) . x) = (( 'not' ((a 'or' c) . x)) 'or' ((b 'or' d) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) 'or' (c . x))) 'or' ((b 'or' d) . x)) by BVFUNC_1:def 4

        .= (((b . x) 'or' (d . x)) 'or' (( 'not' (a . x)) '&' ( 'not' (c . x)))) by BVFUNC_1:def 4

        .= ((((d . x) 'or' (b . x)) 'or' ( 'not' (a . x))) '&' (((b . x) 'or' (d . x)) 'or' ( 'not' (c . x)))) by XBOOLEAN: 9

        .= (((d . x) 'or' ((b . x) 'or' ( 'not' (a . x)))) '&' (((b . x) 'or' (d . x)) 'or' ( 'not' (c . x))))

        .= ( TRUE '&' ((b . x) 'or' TRUE )) by A3, A4, BINARITH: 11

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:23

    for a,b be Function of Y, BOOLEAN holds ((a '&' ( 'not' b)) 'imp' ( 'not' a)) = ( I_el Y) implies (a 'imp' b) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      assume

       A1: ((a '&' ( 'not' b)) 'imp' ( 'not' a)) = ( I_el Y);

      for x be Element of Y holds ((a 'imp' b) . x) = TRUE

      proof

        let x be Element of Y;

        (((a '&' ( 'not' b)) 'imp' ( 'not' a)) . x) = TRUE by A1, BVFUNC_1:def 11;

        then (( 'not' ((a '&' ( 'not' b)) . x)) 'or' (( 'not' a) . x)) = TRUE by BVFUNC_1:def 8;

        then (( 'not' ((a . x) '&' (( 'not' b) . x))) 'or' (( 'not' a) . x)) = TRUE by MARGREL1:def 20;

        then ((( 'not' (a . x)) 'or' ( 'not' ( 'not' (b . x)))) 'or' (( 'not' a) . x)) = TRUE by MARGREL1:def 19;

        then ((( 'not' (a . x)) 'or' (b . x)) 'or' ( 'not' (a . x))) = TRUE by MARGREL1:def 19;

        then ((b . x) 'or' (( 'not' (a . x)) 'or' ( 'not' (a . x)))) = TRUE by XBOOLEAN: 4;

        hence thesis by BVFUNC_1:def 8;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:24

    for a,b be Function of Y, BOOLEAN holds (a 'imp' ( 'not' b)) = ( I_el Y) implies (b 'imp' ( 'not' a)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      assume

       A1: (a 'imp' ( 'not' b)) = ( I_el Y);

      for x be Element of Y holds ((b 'imp' ( 'not' a)) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' ( 'not' b)) . x) = TRUE by A1, BVFUNC_1:def 11;

        then (( 'not' (a . x)) 'or' (( 'not' b) . x)) = TRUE by BVFUNC_1:def 8;

        then

         A2: (( 'not' (a . x)) 'or' ( 'not' (b . x))) = TRUE by MARGREL1:def 19;

        ((b 'imp' ( 'not' a)) . x) = (( 'not' (b . x)) 'or' (( 'not' a) . x)) by BVFUNC_1:def 8

        .= TRUE by A2, MARGREL1:def 19;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:25

    for a,b be Function of Y, BOOLEAN holds (( 'not' a) 'imp' b) = ( I_el Y) implies (( 'not' b) 'imp' a) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      assume

       A1: (( 'not' a) 'imp' b) = ( I_el Y);

      for x be Element of Y holds ((( 'not' b) 'imp' a) . x) = TRUE

      proof

        let x be Element of Y;

        ((( 'not' a) 'imp' b) . x) = TRUE by A1, BVFUNC_1:def 11;

        then (( 'not' (( 'not' a) . x)) 'or' (b . x)) = TRUE by BVFUNC_1:def 8;

        then

         A2: (( 'not' ( 'not' (a . x))) 'or' (b . x)) = TRUE by MARGREL1:def 19;

        ((( 'not' b) 'imp' a) . x) = (( 'not' (( 'not' b) . x)) 'or' (a . x)) by BVFUNC_1:def 8

        .= TRUE by A2, MARGREL1:def 19;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:26



     Th26: for a,b be Function of Y, BOOLEAN holds (a 'imp' (a 'or' b)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((a 'imp' (a 'or' b)) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' (a 'or' b)) . x) = (( 'not' (a . x)) 'or' ((a 'or' b) . x)) by BVFUNC_1:def 8

        .= (( 'not' (a . x)) 'or' ((a . x) 'or' (b . x))) by BVFUNC_1:def 4

        .= ((( 'not' (a . x)) 'or' (a . x)) 'or' (b . x))

        .= ( TRUE 'or' (b . x)) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:27

    for a,b be Function of Y, BOOLEAN holds ((a 'or' b) 'imp' (( 'not' a) 'imp' b)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'or' b) 'imp' (( 'not' a) 'imp' b)) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'or' b) 'imp' (( 'not' a) 'imp' b)) . x) = (( 'not' ((a 'or' b) . x)) 'or' ((( 'not' a) 'imp' b) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) 'or' (b . x))) 'or' ((( 'not' a) 'imp' b) . x)) by BVFUNC_1:def 4

        .= ((( 'not' (a . x)) '&' ( 'not' (b . x))) 'or' (( 'not' (( 'not' a) . x)) 'or' (b . x))) by BVFUNC_1:def 8

        .= (((a . x) 'or' (b . x)) 'or' (( 'not' (a . x)) '&' ( 'not' (b . x)))) by MARGREL1:def 19

        .= ((((a . x) 'or' (b . x)) 'or' ( 'not' (a . x))) '&' (((a . x) 'or' (b . x)) 'or' ( 'not' (b . x)))) by XBOOLEAN: 9

        .= ((((a . x) 'or' ( 'not' (a . x))) 'or' (b . x)) '&' ((a . x) 'or' ((b . x) 'or' ( 'not' (b . x)))))

        .= (( TRUE 'or' (b . x)) '&' ((a . x) 'or' ((b . x) 'or' ( 'not' (b . x))))) by XBOOLEAN: 102

        .= ( TRUE '&' ((a . x) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:28



     Th28: for a,b be Function of Y, BOOLEAN holds (( 'not' (a 'or' b)) 'imp' (( 'not' a) '&' ( 'not' b))) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((( 'not' (a 'or' b)) 'imp' (( 'not' a) '&' ( 'not' b))) . x) = TRUE

      proof

        let x be Element of Y;

        ((( 'not' (a 'or' b)) 'imp' (( 'not' a) '&' ( 'not' b))) . x) = (( 'not' (( 'not' (a 'or' b)) . x)) 'or' ((( 'not' a) '&' ( 'not' b)) . x)) by BVFUNC_1:def 8

        .= (((a 'or' b) . x) 'or' ((( 'not' a) '&' ( 'not' b)) . x)) by MARGREL1:def 19

        .= (((a . x) 'or' (b . x)) 'or' ((( 'not' a) '&' ( 'not' b)) . x)) by BVFUNC_1:def 4

        .= (((a . x) 'or' (b . x)) 'or' ((( 'not' a) . x) '&' (( 'not' b) . x))) by MARGREL1:def 20

        .= (((a . x) 'or' (b . x)) 'or' (( 'not' (a . x)) '&' (( 'not' b) . x))) by MARGREL1:def 19

        .= (((a . x) 'or' (b . x)) 'or' (( 'not' (a . x)) '&' ( 'not' (b . x)))) by MARGREL1:def 19

        .= ((((a . x) 'or' (b . x)) 'or' ( 'not' (a . x))) '&' (((a . x) 'or' (b . x)) 'or' ( 'not' (b . x)))) by XBOOLEAN: 9

        .= ((((a . x) 'or' ( 'not' (a . x))) 'or' (b . x)) '&' ((a . x) 'or' ((b . x) 'or' ( 'not' (b . x)))))

        .= (( TRUE 'or' (b . x)) '&' ((a . x) 'or' ((b . x) 'or' ( 'not' (b . x))))) by XBOOLEAN: 102

        .= ( TRUE '&' ((a . x) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:29

    for a,b be Function of Y, BOOLEAN holds ((( 'not' a) '&' ( 'not' b)) 'imp' ( 'not' (a 'or' b))) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;



      thus ((( 'not' a) '&' ( 'not' b)) 'imp' ( 'not' (a 'or' b))) = (( 'not' (a 'or' b)) 'imp' ( 'not' (a 'or' b))) by BVFUNC_1: 13

      .= (( 'not' (a 'or' b)) 'imp' (( 'not' a) '&' ( 'not' b))) by BVFUNC_1: 13

      .= ( I_el Y) by Th28;

    end;

    theorem :: BVFUNC_6:30

    for a,b be Function of Y, BOOLEAN holds (( 'not' (a 'or' b)) 'imp' ( 'not' a)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((( 'not' (a 'or' b)) 'imp' ( 'not' a)) . x) = TRUE

      proof

        let x be Element of Y;

        ((( 'not' (a 'or' b)) 'imp' ( 'not' a)) . x) = (( 'not' (( 'not' (a 'or' b)) . x)) 'or' (( 'not' a) . x)) by BVFUNC_1:def 8

        .= (( 'not' ( 'not' ((a 'or' b) . x))) 'or' (( 'not' a) . x)) by MARGREL1:def 19

        .= (((a 'or' b) . x) 'or' ( 'not' (a . x))) by MARGREL1:def 19

        .= (((a . x) 'or' (b . x)) 'or' ( 'not' (a . x))) by BVFUNC_1:def 4

        .= (((a . x) 'or' ( 'not' (a . x))) 'or' (b . x))

        .= ( TRUE 'or' (b . x)) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:31

    for a be Function of Y, BOOLEAN holds ((a 'or' a) 'imp' a) = ( I_el Y)

    proof

      let a be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'or' a) 'imp' a) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'or' a) 'imp' a) . x) = (( 'not' (a . x)) 'or' (a . x)) by BVFUNC_1:def 8

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:32

    for a,b be Function of Y, BOOLEAN holds ((a '&' ( 'not' a)) 'imp' b) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a '&' ( 'not' a)) 'imp' b) . x) = TRUE

      proof

        let x be Element of Y;

        (((a '&' ( 'not' a)) 'imp' b) . x) = (( 'not' ((a '&' ( 'not' a)) . x)) 'or' (b . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) '&' (( 'not' a) . x))) 'or' (b . x)) by MARGREL1:def 20

        .= ((( 'not' (a . x)) 'or' ( 'not' ( 'not' (a . x)))) 'or' (b . x)) by MARGREL1:def 19

        .= ( TRUE 'or' (b . x)) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:33

    for a,b be Function of Y, BOOLEAN holds ((a 'imp' b) 'imp' (( 'not' a) 'or' b)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'imp' b) 'imp' (( 'not' a) 'or' b)) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'imp' b) 'imp' (( 'not' a) 'or' b)) . x) = (( 'not' ((a 'imp' b) . x)) 'or' ((( 'not' a) 'or' b) . x)) by BVFUNC_1:def 8

        .= (( 'not' (( 'not' (a . x)) 'or' (b . x))) 'or' ((( 'not' a) 'or' b) . x)) by BVFUNC_1:def 8

        .= ((( 'not' ( 'not' (a . x))) '&' ( 'not' (b . x))) 'or' ((( 'not' a) . x) 'or' (b . x))) by BVFUNC_1:def 4

        .= ((( 'not' (a . x)) 'or' (b . x)) 'or' ((a . x) '&' ( 'not' (b . x)))) by MARGREL1:def 19

        .= (((( 'not' (a . x)) 'or' (b . x)) 'or' (a . x)) '&' ((( 'not' (a . x)) 'or' (b . x)) 'or' ( 'not' (b . x)))) by XBOOLEAN: 9

        .= (((( 'not' (a . x)) 'or' (a . x)) 'or' (b . x)) '&' (( 'not' (a . x)) 'or' ((b . x) 'or' ( 'not' (b . x)))))

        .= (( TRUE 'or' (b . x)) '&' (( 'not' (a . x)) 'or' ((b . x) 'or' ( 'not' (b . x))))) by XBOOLEAN: 102

        .= ( TRUE '&' (( 'not' (a . x)) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:34

    for a,b be Function of Y, BOOLEAN holds ((a '&' b) 'imp' ( 'not' (a 'imp' ( 'not' b)))) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a '&' b) 'imp' ( 'not' (a 'imp' ( 'not' b)))) . x) = TRUE

      proof

        let x be Element of Y;

        (((a '&' b) 'imp' ( 'not' (a 'imp' ( 'not' b)))) . x) = (( 'not' ((a '&' b) . x)) 'or' (( 'not' (a 'imp' ( 'not' b))) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) '&' (b . x))) 'or' (( 'not' (a 'imp' ( 'not' b))) . x)) by MARGREL1:def 20

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ( 'not' ((a 'imp' ( 'not' b)) . x))) by MARGREL1:def 19

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ( 'not' (( 'not' (a . x)) 'or' (( 'not' b) . x)))) by BVFUNC_1:def 8

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ((a . x) '&' (b . x))) by MARGREL1:def 19

        .= (((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (a . x)) '&' ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (b . x))) by XBOOLEAN: 9

        .= (((( 'not' (a . x)) 'or' (a . x)) 'or' ( 'not' (b . x))) '&' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (b . x))))

        .= (( TRUE 'or' ( 'not' (b . x))) '&' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (b . x)))) by XBOOLEAN: 102

        .= ( TRUE '&' (( 'not' (a . x)) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:35

    for a,b be Function of Y, BOOLEAN holds (( 'not' (a 'imp' ( 'not' b))) 'imp' (a '&' b)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((( 'not' (a 'imp' ( 'not' b))) 'imp' (a '&' b)) . x) = TRUE

      proof

        let x be Element of Y;

        ((( 'not' (a 'imp' ( 'not' b))) 'imp' (a '&' b)) . x) = (( 'not' (( 'not' (a 'imp' ( 'not' b))) . x)) 'or' ((a '&' b) . x)) by BVFUNC_1:def 8

        .= (( 'not' ( 'not' ((a 'imp' ( 'not' b)) . x))) 'or' ((a '&' b) . x)) by MARGREL1:def 19

        .= (( 'not' ( 'not' ((a 'imp' ( 'not' b)) . x))) 'or' ((a . x) '&' (b . x))) by MARGREL1:def 20

        .= ((( 'not' (a . x)) 'or' (( 'not' b) . x)) 'or' ((a . x) '&' (b . x))) by BVFUNC_1:def 8

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ((a . x) '&' (b . x))) by MARGREL1:def 19

        .= (((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (a . x)) '&' ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (b . x))) by XBOOLEAN: 9

        .= (((( 'not' (a . x)) 'or' (a . x)) 'or' ( 'not' (b . x))) '&' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (b . x))))

        .= (( TRUE 'or' ( 'not' (b . x))) '&' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (b . x)))) by XBOOLEAN: 102

        .= ( TRUE '&' (( 'not' (a . x)) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:36



     Th36: for a,b be Function of Y, BOOLEAN holds (( 'not' (a '&' b)) 'imp' (( 'not' a) 'or' ( 'not' b))) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((( 'not' (a '&' b)) 'imp' (( 'not' a) 'or' ( 'not' b))) . x) = TRUE

      proof

        let x be Element of Y;

        ((( 'not' (a '&' b)) 'imp' (( 'not' a) 'or' ( 'not' b))) . x) = (( 'not' (( 'not' (a '&' b)) . x)) 'or' ((( 'not' a) 'or' ( 'not' b)) . x)) by BVFUNC_1:def 8

        .= (((a '&' b) . x) 'or' ((( 'not' a) 'or' ( 'not' b)) . x)) by MARGREL1:def 19

        .= (((a . x) '&' (b . x)) 'or' ((( 'not' a) 'or' ( 'not' b)) . x)) by MARGREL1:def 20

        .= (((( 'not' a) . x) 'or' (( 'not' b) . x)) 'or' ((a . x) '&' (b . x))) by BVFUNC_1:def 4

        .= ((( 'not' (a . x)) 'or' (( 'not' b) . x)) 'or' ((a . x) '&' (b . x))) by MARGREL1:def 19

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ((a . x) '&' (b . x))) by MARGREL1:def 19

        .= (((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (a . x)) '&' ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (b . x))) by XBOOLEAN: 9

        .= (((( 'not' (a . x)) 'or' (a . x)) 'or' ( 'not' (b . x))) '&' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (b . x))))

        .= (( TRUE 'or' ( 'not' (b . x))) '&' (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (b . x)))) by XBOOLEAN: 102

        .= ( TRUE '&' (( 'not' (a . x)) 'or' TRUE )) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:37

    for a,b be Function of Y, BOOLEAN holds ((( 'not' a) 'or' ( 'not' b)) 'imp' ( 'not' (a '&' b))) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;



      thus ((( 'not' a) 'or' ( 'not' b)) 'imp' ( 'not' (a '&' b))) = (( 'not' (a '&' b)) 'imp' ( 'not' (a '&' b))) by BVFUNC_1: 14

      .= (( 'not' (a '&' b)) 'imp' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 14

      .= ( I_el Y) by Th36;

    end;

    theorem :: BVFUNC_6:38



     Th38: for a,b be Function of Y, BOOLEAN holds ((a '&' b) 'imp' a) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a '&' b) 'imp' a) . x) = TRUE

      proof

        let x be Element of Y;

        (((a '&' b) 'imp' a) . x) = (( 'not' ((a '&' b) . x)) 'or' (a . x)) by BVFUNC_1:def 8

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (a . x)) by MARGREL1:def 20

        .= ((( 'not' (a . x)) 'or' (a . x)) 'or' ( 'not' (b . x)))

        .= ( TRUE 'or' ( 'not' (b . x))) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:39

    for a,b be Function of Y, BOOLEAN holds ((a '&' b) 'imp' (a 'or' b)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a '&' b) 'imp' (a 'or' b)) . x) = TRUE

      proof

        let x be Element of Y;

        (((a '&' b) 'imp' (a 'or' b)) . x) = (( 'not' ((a '&' b) . x)) 'or' ((a 'or' b) . x)) by BVFUNC_1:def 8

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ((a 'or' b) . x)) by MARGREL1:def 20

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ((a . x) 'or' (b . x))) by BVFUNC_1:def 4

        .= ((( 'not' (b . x)) 'or' (( 'not' (a . x)) 'or' (a . x))) 'or' (b . x))

        .= ((( 'not' (b . x)) 'or' TRUE ) 'or' (b . x)) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:40

    for a,b be Function of Y, BOOLEAN holds ((a '&' b) 'imp' b) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a '&' b) 'imp' b) . x) = TRUE

      proof

        let x be Element of Y;

        (((a '&' b) 'imp' b) . x) = (( 'not' ((a '&' b) . x)) 'or' (b . x)) by BVFUNC_1:def 8

        .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (b . x)) by MARGREL1:def 20

        .= (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (b . x)))

        .= (( 'not' (a . x)) 'or' TRUE ) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:41

    for a be Function of Y, BOOLEAN holds (a 'imp' (a '&' a)) = ( I_el Y)

    proof

      let a be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((a 'imp' (a '&' a)) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' (a '&' a)) . x) = ( TRUE '&' (( 'not' (a . x)) 'or' (a . x))) by BVFUNC_1:def 8

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:42



     Th42: for a,b be Function of Y, BOOLEAN holds ((a 'eqv' b) 'imp' (a 'imp' b)) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'eqv' b) 'imp' (a 'imp' b)) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'eqv' b) 'imp' (a 'imp' b)) . x) = (( 'not' ((a 'eqv' b) . x)) 'or' ((a 'imp' b) . x)) by BVFUNC_1:def 8

        .= (((a . x) 'xor' (b . x)) 'or' ((a 'imp' b) . x)) by BVFUNC_1:def 9

        .= (((( 'not' (a . x)) '&' (b . x)) 'or' ((a . x) '&' ( 'not' (b . x)))) 'or' ( 'not' ((a . x) '&' ( 'not' (b . x))))) by BVFUNC_1:def 8

        .= ((( 'not' (a . x)) '&' (b . x)) 'or' (((a . x) '&' ( 'not' (b . x))) 'or' ( 'not' ((a . x) '&' ( 'not' (b . x))))))

        .= ((( 'not' (a . x)) '&' (b . x)) 'or' TRUE ) by XBOOLEAN: 102

        .= TRUE ;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:43

    for a,b be Function of Y, BOOLEAN holds ((a 'eqv' b) 'imp' (b 'imp' a)) = ( I_el Y) by Th42;

    theorem :: BVFUNC_6:44

    for a,b,c be Function of Y, BOOLEAN holds (((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))) . x) = (( 'not' (((a 'or' b) 'or' c) . x)) 'or' ((a 'or' (b 'or' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (((a 'or' b) . x) 'or' (c . x))) 'or' ((a 'or' (b 'or' c)) . x)) by BVFUNC_1:def 4

        .= (( 'not' (((a . x) 'or' (b . x)) 'or' (c . x))) 'or' ((a 'or' (b 'or' c)) . x)) by BVFUNC_1:def 4

        .= (( 'not' (((a . x) 'or' (b . x)) 'or' (c . x))) 'or' ((a . x) 'or' ((b 'or' c) . x))) by BVFUNC_1:def 4

        .= (( 'not' (((a . x) 'or' (b . x)) 'or' (c . x))) 'or' ((a . x) 'or' ((b . x) 'or' (c . x)))) by BVFUNC_1:def 4

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:45

    for a,b,c be Function of Y, BOOLEAN holds (((a '&' b) '&' c) 'imp' (a '&' (b '&' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a '&' b) '&' c) 'imp' (a '&' (b '&' c))) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a '&' b) '&' c) 'imp' (a '&' (b '&' c))) . x) = (( 'not' (((a '&' b) '&' c) . x)) 'or' ((a '&' (b '&' c)) . x)) by BVFUNC_1:def 8

        .= (( 'not' (((a '&' b) . x) '&' (c . x))) 'or' ((a '&' (b '&' c)) . x)) by MARGREL1:def 20

        .= (( 'not' (((a . x) '&' (b . x)) '&' (c . x))) 'or' ((a '&' (b '&' c)) . x)) by MARGREL1:def 20

        .= (( 'not' (((a . x) '&' (b . x)) '&' (c . x))) 'or' ((a . x) '&' ((b '&' c) . x))) by MARGREL1:def 20

        .= (( 'not' (((a . x) '&' (b . x)) '&' (c . x))) 'or' ((a . x) '&' ((b . x) '&' (c . x)))) by MARGREL1:def 20

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:46

    for a,b,c be Function of Y, BOOLEAN holds ((a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)) . x) = TRUE

      proof

        let x be Element of Y;

        (((a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)) . x) = (( 'not' ((a 'or' (b 'or' c)) . x)) 'or' (((a 'or' b) 'or' c) . x)) by BVFUNC_1:def 8

        .= (( 'not' ((a . x) 'or' ((b 'or' c) . x))) 'or' (((a 'or' b) 'or' c) . x)) by BVFUNC_1:def 4

        .= (( 'not' ((a . x) 'or' ((b . x) 'or' (c . x)))) 'or' (((a 'or' b) 'or' c) . x)) by BVFUNC_1:def 4

        .= (( 'not' ((a . x) 'or' ((b . x) 'or' (c . x)))) 'or' (((a 'or' b) . x) 'or' (c . x))) by BVFUNC_1:def 4

        .= (( 'not' ((a . x) 'or' ((b . x) 'or' (c . x)))) 'or' (((a . x) 'or' (b . x)) 'or' (c . x))) by BVFUNC_1:def 4

        .= TRUE by XBOOLEAN: 102;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    begin

    reserve Y for non empty set;

    theorem :: BVFUNC_6:47

    for a,b be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (( 'not' a) 'imp' b)) = b

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;



       A1: (((a 'imp' b) '&' (( 'not' a) 'imp' b)) . x) = (((a 'imp' b) . x) '&' ((( 'not' a) 'imp' b) . x)) by MARGREL1:def 20

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' ((( 'not' a) 'imp' b) . x)) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (( 'not' a) . x)) 'or' (b . x))) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' ((a . x) 'or' (b . x))) by MARGREL1:def 19;

      now

        per cases by XBOOLEAN:def 3;

          case (a . x) = TRUE ;



          then (((a 'imp' b) '&' (( 'not' a) 'imp' b)) . x) = (( FALSE 'or' (b . x)) '&' ( TRUE 'or' (b . x))) by A1

          .= (( FALSE 'or' (b . x)) '&' TRUE )

          .= ( TRUE '&' (b . x))

          .= (b . x);

          hence thesis;

        end;

          case (a . x) = FALSE ;



          then (((a 'imp' b) '&' (( 'not' a) 'imp' b)) . x) = (( TRUE 'or' (b . x)) '&' ( FALSE 'or' (b . x))) by A1

          .= ( TRUE '&' ( FALSE 'or' (b . x)))

          .= ( TRUE '&' (b . x))

          .= (b . x);

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:48

    for a,b be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (a 'imp' ( 'not' b))) = ( 'not' a)

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;



       A1: (((a 'imp' b) '&' (a 'imp' ( 'not' b))) . x) = (((a 'imp' b) . x) '&' ((a 'imp' ( 'not' b)) . x)) by MARGREL1:def 20

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' ((a 'imp' ( 'not' b)) . x)) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (a . x)) 'or' (( 'not' b) . x))) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (a . x)) 'or' ( 'not' (b . x)))) by MARGREL1:def 19;

      now

        per cases by XBOOLEAN:def 3;

          case (b . x) = TRUE ;



          then (((a 'imp' b) '&' (a 'imp' ( 'not' b))) . x) = ((( 'not' (a . x)) 'or' TRUE ) '&' (( 'not' (a . x)) 'or' FALSE )) by A1

          .= ((( 'not' (a . x)) 'or' TRUE ) '&' ( 'not' (a . x)))

          .= ( TRUE '&' ( 'not' (a . x)))

          .= ( 'not' (a . x))

          .= (( 'not' a) . x) by MARGREL1:def 19;

          hence thesis;

        end;

          case (b . x) = FALSE ;



          then (((a 'imp' b) '&' (a 'imp' ( 'not' b))) . x) = ((( 'not' (a . x)) 'or' FALSE ) '&' (( 'not' (a . x)) 'or' TRUE )) by A1

          .= (( 'not' (a . x)) '&' (( 'not' (a . x)) 'or' TRUE ))

          .= ( TRUE '&' ( 'not' (a . x)))

          .= ( 'not' (a . x))

          .= (( 'not' a) . x) by MARGREL1:def 19;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:49



     Th73: for a,b,c be Function of Y, BOOLEAN holds (a 'imp' (b 'or' c)) = ((a 'imp' b) 'or' (a 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'imp' b) 'or' (a 'imp' c)) . x) = (((a 'imp' b) . x) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 4

      .= ((( 'not' (a . x)) 'or' (b . x)) 'or' ((a 'imp' c) . x)) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (b . x)) 'or' (( 'not' (a . x)) 'or' (c . x))) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (( 'not' (a . x)) 'or' (b . x))) 'or' (c . x))

      .= (((( 'not' (a . x)) 'or' ( 'not' (a . x))) 'or' (b . x)) 'or' (c . x)) by BINARITH: 11

      .= (( 'not' (a . x)) 'or' ((b . x) 'or' (c . x)))

      .= (( 'not' (a . x)) 'or' ((b 'or' c) . x)) by BVFUNC_1:def 4

      .= ((a 'imp' (b 'or' c)) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:50

    for a,b,c be Function of Y, BOOLEAN holds (a 'imp' (b '&' c)) = ((a 'imp' b) '&' (a 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'imp' b) '&' (a 'imp' c)) . x) = (((a 'imp' b) . x) '&' ((a 'imp' c) . x)) by MARGREL1:def 20

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' ((a 'imp' c) . x)) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (a . x)) 'or' (c . x))) by BVFUNC_1:def 8

      .= (( 'not' (a . x)) 'or' ((b . x) '&' (c . x))) by XBOOLEAN: 9

      .= (( 'not' (a . x)) 'or' ((b '&' c) . x)) by MARGREL1:def 20

      .= ((a 'imp' (b '&' c)) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:51



     Th75: for a,b,c be Function of Y, BOOLEAN holds ((a 'or' b) 'imp' c) = ((a 'imp' c) '&' (b 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'imp' c) '&' (b 'imp' c)) . x) = (((a 'imp' c) . x) '&' ((b 'imp' c) . x)) by MARGREL1:def 20

      .= ((( 'not' (a . x)) 'or' (c . x)) '&' ((b 'imp' c) . x)) by BVFUNC_1:def 8

      .= (((c . x) 'or' ( 'not' (a . x))) '&' (( 'not' (b . x)) 'or' (c . x))) by BVFUNC_1:def 8

      .= (( 'not' ((a . x) 'or' (b . x))) 'or' (c . x)) by XBOOLEAN: 9

      .= (( 'not' ((a 'or' b) . x)) 'or' (c . x)) by BVFUNC_1:def 4

      .= (((a 'or' b) 'imp' c) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:52



     Th76: for a,b,c be Function of Y, BOOLEAN holds ((a '&' b) 'imp' c) = ((a 'imp' c) 'or' (b 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'imp' c) 'or' (b 'imp' c)) . x) = (((a 'imp' c) . x) 'or' ((b 'imp' c) . x)) by BVFUNC_1:def 4

      .= ((( 'not' (a . x)) 'or' (c . x)) 'or' ((b 'imp' c) . x)) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (c . x)) 'or' (( 'not' (b . x)) 'or' (c . x))) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' ((c . x) 'or' ( 'not' (b . x)))) 'or' (c . x))

      .= (((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (c . x)) 'or' (c . x))

      .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ((c . x) 'or' (c . x))) by BINARITH: 11

      .= (( 'not' ((a '&' b) . x)) 'or' (c . x)) by MARGREL1:def 20

      .= (((a '&' b) 'imp' c) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:53



     Th7: for a,b,c be Function of Y, BOOLEAN holds ((a '&' b) 'imp' c) = (a 'imp' (b 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'imp' (b 'imp' c)) . x) = (( 'not' (a . x)) 'or' ((b 'imp' c) . x)) by BVFUNC_1:def 8

      .= (( 'not' (a . x)) 'or' (( 'not' (b . x)) 'or' (c . x))) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (c . x))

      .= (( 'not' ((a '&' b) . x)) 'or' (c . x)) by MARGREL1:def 20

      .= (((a '&' b) 'imp' c) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:54

    for a,b,c be Function of Y, BOOLEAN holds ((a '&' b) 'imp' c) = (a 'imp' (( 'not' b) 'or' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'imp' (( 'not' b) 'or' c)) . x) = ((a 'imp' (b 'imp' c)) . x) by BVFUNC_4: 8;

      hence thesis by Th7;

    end;

    theorem :: BVFUNC_6:55

    for a,b,c be Function of Y, BOOLEAN holds (a 'imp' (b 'or' c)) = ((a '&' ( 'not' b)) 'imp' c)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a '&' ( 'not' b)) 'imp' c) . x) = (( 'not' ((a '&' ( 'not' b)) . x)) 'or' (c . x)) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' ( 'not' (( 'not' b) . x))) 'or' (c . x)) by MARGREL1:def 20

      .= ((( 'not' (a . x)) 'or' (b . x)) 'or' (c . x)) by MARGREL1:def 19

      .= (( 'not' (a . x)) 'or' ((b . x) 'or' (c . x)))

      .= (( 'not' (a . x)) 'or' ((b 'or' c) . x)) by BVFUNC_1:def 4

      .= ((a 'imp' (b 'or' c)) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:56

    for a,b be Function of Y, BOOLEAN holds (a '&' (a 'imp' b)) = (a '&' b)

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a '&' (a 'imp' b)) . x) = ((a . x) '&' ((a 'imp' b) . x)) by MARGREL1:def 20

      .= ((a . x) '&' (( 'not' (a . x)) 'or' (b . x))) by BVFUNC_1:def 8

      .= (((a . x) '&' ( 'not' (a . x))) 'or' ((a . x) '&' (b . x))) by XBOOLEAN: 8

      .= ( FALSE 'or' ((a . x) '&' (b . x))) by XBOOLEAN: 138

      .= ((a . x) '&' (b . x))

      .= ((a '&' b) . x) by MARGREL1:def 20;

      hence thesis;

    end;

    theorem :: BVFUNC_6:57

    for a,b be Function of Y, BOOLEAN holds ((a 'imp' b) '&' ( 'not' b)) = (( 'not' a) '&' ( 'not' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'imp' b) '&' ( 'not' b)) . x) = (((a 'imp' b) . x) '&' (( 'not' b) . x)) by MARGREL1:def 20

      .= ((( 'not' b) . x) '&' (( 'not' (a . x)) 'or' (b . x))) by BVFUNC_1:def 8

      .= (((( 'not' b) . x) '&' ( 'not' (a . x))) 'or' ((( 'not' b) . x) '&' (b . x))) by XBOOLEAN: 8

      .= (((( 'not' b) . x) '&' ( 'not' (a . x))) 'or' ((b . x) '&' ( 'not' (b . x)))) by MARGREL1:def 19

      .= (((( 'not' b) . x) '&' ( 'not' (a . x))) 'or' FALSE ) by XBOOLEAN: 138

      .= ((( 'not' b) . x) '&' ( 'not' (a . x)))

      .= ((( 'not' b) . x) '&' (( 'not' a) . x)) by MARGREL1:def 19

      .= ((( 'not' a) '&' ( 'not' b)) . x) by MARGREL1:def 20;

      hence thesis;

    end;

    theorem :: BVFUNC_6:58



     Th12: for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) = (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;



       A1: ((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x) = ((((a 'imp' b) '&' (b 'imp' c)) . x) '&' ((a 'imp' c) . x)) by MARGREL1:def 20

      .= ((((a 'imp' b) . x) '&' ((b 'imp' c) . x)) '&' ((a 'imp' c) . x)) by MARGREL1:def 20

      .= (((( 'not' (a . x)) 'or' (b . x)) '&' ((b 'imp' c) . x)) '&' ((a 'imp' c) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' ((a 'imp' c) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' (( 'not' (a . x)) 'or' (c . x))) by BVFUNC_1:def 8

      .= ((((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' ( 'not' (a . x))) 'or' (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' (c . x))) by XBOOLEAN: 8;



       A2: ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) = (((a 'imp' b) . x) '&' (( 'not' (b . x)) 'or' (c . x))) by BVFUNC_1:def 8

      .= (((a 'imp' b) . x) '&' ((b 'imp' c) . x)) by BVFUNC_1:def 8

      .= (((a 'imp' b) '&' (b 'imp' c)) . x) by MARGREL1:def 20;



       A3: (((a 'imp' b) '&' (b 'imp' c)) . x) = (((a 'imp' b) . x) '&' ((b 'imp' c) . x)) by MARGREL1:def 20

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' ((b 'imp' c) . x)) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) by BVFUNC_1:def 8;

      now

        per cases by XBOOLEAN:def 3;

          case (a . x) = TRUE & (c . x) = TRUE ;



          then ((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x) = ((((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' FALSE ) 'or' (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' TRUE )) by A1

          .= ( FALSE 'or' (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' TRUE ))

          .= ( FALSE 'or' ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))))

          .= (((a 'imp' b) '&' (b 'imp' c)) . x) by A2;

          hence thesis;

        end;

          case

           A4: (a . x) = TRUE & (c . x) = FALSE ;



          then

           A5: (((a 'imp' b) '&' (b 'imp' c)) . x) = (( FALSE 'or' (b . x)) '&' (( 'not' (b . x)) 'or' FALSE )) by A3

          .= (( FALSE 'or' (b . x)) '&' ( 'not' (b . x)))

          .= ((b . x) '&' ( 'not' (b . x)))

          .= FALSE by XBOOLEAN: 138;

          ((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x) = ((((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' FALSE ) 'or' (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' FALSE )) by A1, A4

          .= FALSE ;

          hence thesis by A5;

        end;

          case (a . x) = FALSE & (c . x) = TRUE ;



          then ((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x) = ((((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' TRUE ) 'or' (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) '&' TRUE )) by A1

          .= (((a 'imp' b) '&' (b 'imp' c)) . x) by A2;

          hence thesis;

        end;

          case (a . x) = FALSE & (c . x) = FALSE ;



          then ((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x) = (( TRUE '&' ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x)))) 'or' ( FALSE '&' ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))))) by A1

          .= (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) 'or' ( FALSE '&' ((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x)))))

          .= (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (b . x)) 'or' (c . x))) 'or' FALSE )

          .= (((a 'imp' b) '&' (b 'imp' c)) . x) by A2;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:59

    for a be Function of Y, BOOLEAN holds (( I_el Y) 'imp' a) = a

    proof

      let a be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((( I_el Y) 'imp' a) . x) = (( 'not' (( I_el Y) . x)) 'or' (a . x)) by BVFUNC_1:def 8

      .= ( FALSE 'or' (a . x)) by BVFUNC_1:def 11;

      hence thesis;

    end;

    theorem :: BVFUNC_6:60

    for a be Function of Y, BOOLEAN holds (a 'imp' ( O_el Y)) = ( 'not' a)

    proof

      let a be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'imp' ( O_el Y)) . x) = (( 'not' (a . x)) 'or' (( O_el Y) . x)) by BVFUNC_1:def 8

      .= (( 'not' (a . x)) 'or' FALSE ) by BVFUNC_1:def 10

      .= ( 'not' (a . x));

      hence thesis by MARGREL1:def 19;

    end;

    theorem :: BVFUNC_6:61

    for a be Function of Y, BOOLEAN holds (( O_el Y) 'imp' a) = ( I_el Y)

    proof

      let a be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((( O_el Y) 'imp' a) . x) = TRUE

      proof

        let x be Element of Y;

        ((( O_el Y) 'imp' a) . x) = (( 'not' (( O_el Y) . x)) 'or' (a . x)) by BVFUNC_1:def 8

        .= ( TRUE 'or' (a . x)) by BVFUNC_1:def 10;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:62

    for a be Function of Y, BOOLEAN holds (a 'imp' ( I_el Y)) = ( I_el Y)

    proof

      let a be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((a 'imp' ( I_el Y)) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' ( I_el Y)) . x) = (( 'not' (a . x)) 'or' (( I_el Y) . x)) by BVFUNC_1:def 8

        .= (( 'not' (a . x)) 'or' TRUE ) by BVFUNC_1:def 11;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:63

    for a be Function of Y, BOOLEAN holds (a 'imp' ( 'not' a)) = ( 'not' a)

    proof

      let a be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'imp' ( 'not' a)) . x) = (( 'not' (a . x)) 'or' (( 'not' a) . x)) by BVFUNC_1:def 8

      .= ((( 'not' a) . x) 'or' (( 'not' a) . x)) by MARGREL1:def 19

      .= (( 'not' a) . x);

      hence thesis;

    end;

    theorem :: BVFUNC_6:64

    for a,b,c be Function of Y, BOOLEAN holds (a 'imp' b) '<' ((c 'imp' a) 'imp' (c 'imp' b))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;

      assume ((a 'imp' b) . z) = TRUE ;

      then

       A1: (( 'not' (a . z)) 'or' (b . z)) = TRUE by BVFUNC_1:def 8;



       A2: (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;

      now

        per cases by A1, A2;

          case

           A3: ( 'not' (a . z)) = TRUE ;

          (((c 'imp' a) 'imp' (c 'imp' b)) . z) = (( 'not' ((c 'imp' a) . z)) 'or' ((c 'imp' b) . z)) by BVFUNC_1:def 8

          .= (( 'not' (( 'not' (c . z)) 'or' (a . z))) 'or' ((c 'imp' b) . z)) by BVFUNC_1:def 8

          .= ((c . z) 'or' ((c 'imp' b) . z)) by A3

          .= ((c . z) 'or' (( 'not' (c . z)) 'or' (b . z))) by BVFUNC_1:def 8

          .= (((c . z) 'or' ( 'not' (c . z))) 'or' (b . z))

          .= ( TRUE 'or' (b . z)) by XBOOLEAN: 102

          .= TRUE ;

          hence thesis;

        end;

          case

           A4: (b . z) = TRUE ;

          (((c 'imp' a) 'imp' (c 'imp' b)) . z) = (( 'not' ((c 'imp' a) . z)) 'or' ((c 'imp' b) . z)) by BVFUNC_1:def 8

          .= (( 'not' (( 'not' (c . z)) 'or' (a . z))) 'or' ((c 'imp' b) . z)) by BVFUNC_1:def 8

          .= (( 'not' (( 'not' (c . z)) 'or' (a . z))) 'or' (( 'not' (c . z)) 'or' TRUE )) by A4, BVFUNC_1:def 8

          .= (( 'not' (( 'not' (c . z)) 'or' (a . z))) 'or' TRUE )

          .= TRUE ;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:65

    for a,b,c be Function of Y, BOOLEAN holds (a 'eqv' b) '<' ((a 'eqv' c) 'eqv' (b 'eqv' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: ((a 'eqv' b) . z) = (((a 'imp' b) '&' (b 'imp' a)) . z) by BVFUNC_4: 7

      .= (((a 'imp' b) . z) '&' ((b 'imp' a) . z)) by MARGREL1:def 20;

      assume

       A2: ((a 'eqv' b) . z) = TRUE ;

      then ((a 'imp' b) . z) = TRUE by A1, MARGREL1: 12;

      then

       A3: (( 'not' (a . z)) 'or' (b . z)) = TRUE by BVFUNC_1:def 8;



       A4: (a . z) = TRUE or (a . z) = FALSE by XBOOLEAN:def 3;



       A5: (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;



       A6: ((b 'imp' a) . z) = (( 'not' (b . z)) 'or' (a . z)) by BVFUNC_1:def 8;



       A7: (((a 'eqv' c) 'eqv' (b 'eqv' c)) . z) = ((((a 'eqv' c) 'imp' (b 'eqv' c)) '&' ((b 'eqv' c) 'imp' (a 'eqv' c))) . z) by BVFUNC_4: 7

      .= (((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'eqv' c)) '&' ((b 'eqv' c) 'imp' (a 'eqv' c))) . z) by BVFUNC_4: 7

      .= (((((a 'imp' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&' ((b 'eqv' c) 'imp' (a 'eqv' c))) . z) by BVFUNC_4: 7

      .= (((((a 'imp' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' (a 'eqv' c))) . z) by BVFUNC_4: 7

      .= (((((a 'imp' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))) . z) by BVFUNC_4: 7

      .= (((((( 'not' a) 'or' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) 'imp' ((( 'not' b) 'or' c) '&' (c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) 'imp' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) 'imp' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (((( 'not' b) 'or' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) 'imp' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) 'imp' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b)) 'imp' ((( 'not' a) 'or' c) '&' (c 'imp' a)))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) 'imp' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b)) 'imp' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_4: 8

      .= (((( 'not' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b)) 'imp' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_4: 8

      .= (((( 'not' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (( 'not' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_4: 8

      .= ((((( 'not' (( 'not' a) 'or' c)) 'or' ( 'not' (( 'not' c) 'or' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (( 'not' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_1: 14

      .= ((((( 'not' (( 'not' a) 'or' c)) 'or' ( 'not' (( 'not' c) 'or' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' ((( 'not' (( 'not' b) 'or' c)) 'or' ( 'not' (( 'not' c) 'or' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_1: 14

      .= (((((( 'not' ( 'not' a)) '&' ( 'not' c)) 'or' ( 'not' (( 'not' c) 'or' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' ((( 'not' (( 'not' b) 'or' c)) 'or' ( 'not' (( 'not' c) 'or' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_1: 13

      .= (((((( 'not' ( 'not' a)) '&' ( 'not' c)) 'or' (( 'not' ( 'not' c)) '&' ( 'not' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' ((( 'not' (( 'not' b) 'or' c)) 'or' ( 'not' (( 'not' c) 'or' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_1: 13

      .= (((((( 'not' ( 'not' a)) '&' ( 'not' c)) 'or' (( 'not' ( 'not' c)) '&' ( 'not' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (((( 'not' ( 'not' b)) '&' ( 'not' c)) 'or' ( 'not' (( 'not' c) 'or' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_1: 13

      .= (((((a '&' ( 'not' c)) 'or' (c '&' ( 'not' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) '&' (((b '&' ( 'not' c)) 'or' (( 'not' ( 'not' c)) '&' ( 'not' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)))) . z) by BVFUNC_1: 13

      .= (((((a '&' ( 'not' c)) 'or' (c '&' ( 'not' a))) 'or' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b))) . z) '&' ((((b '&' ( 'not' c)) 'or' (c '&' ( 'not' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a))) . z)) by MARGREL1:def 20

      .= (((((a '&' ( 'not' c)) 'or' (c '&' ( 'not' a))) . z) 'or' (((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b)) . z)) '&' ((((b '&' ( 'not' c)) 'or' (c '&' ( 'not' b))) 'or' ((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a))) . z)) by BVFUNC_1:def 4

      .= (((((a '&' ( 'not' c)) 'or' (c '&' ( 'not' a))) . z) 'or' (((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b)) . z)) '&' ((((b '&' ( 'not' c)) 'or' (c '&' ( 'not' b))) . z) 'or' (((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) . z))) by BVFUNC_1:def 4

      .= (((((a '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' a)) . z)) 'or' (((( 'not' b) 'or' c) '&' (( 'not' c) 'or' b)) . z)) '&' ((((b '&' ( 'not' c)) 'or' (c '&' ( 'not' b))) . z) 'or' (((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) . z))) by BVFUNC_1:def 4

      .= (((((a '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' a)) . z)) 'or' (((( 'not' b) 'or' c) . z) '&' ((( 'not' c) 'or' b) . z))) '&' ((((b '&' ( 'not' c)) 'or' (c '&' ( 'not' b))) . z) 'or' (((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) . z))) by MARGREL1:def 20

      .= (((((a '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' a)) . z)) 'or' (((( 'not' b) 'or' c) . z) '&' ((( 'not' c) 'or' b) . z))) '&' ((((b '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' b)) . z)) 'or' (((( 'not' a) 'or' c) '&' (( 'not' c) 'or' a)) . z))) by BVFUNC_1:def 4

      .= (((((a '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' a)) . z)) 'or' (((( 'not' b) 'or' c) . z) '&' ((( 'not' c) 'or' b) . z))) '&' ((((b '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' b)) . z)) 'or' (((( 'not' a) 'or' c) . z) '&' ((( 'not' c) 'or' a) . z)))) by MARGREL1:def 20

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c '&' ( 'not' a)) . z)) 'or' (((( 'not' b) 'or' c) . z) '&' ((( 'not' c) 'or' b) . z))) '&' ((((b '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' b)) . z)) 'or' (((( 'not' a) 'or' c) . z) '&' ((( 'not' c) 'or' a) . z)))) by MARGREL1:def 20

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' a) . z))) 'or' (((( 'not' b) 'or' c) . z) '&' ((( 'not' c) 'or' b) . z))) '&' ((((b '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' b)) . z)) 'or' (((( 'not' a) 'or' c) . z) '&' ((( 'not' c) 'or' a) . z)))) by MARGREL1:def 20

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' a) . z))) 'or' (((( 'not' b) . z) 'or' (c . z)) '&' ((( 'not' c) 'or' b) . z))) '&' ((((b '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' b)) . z)) 'or' (((( 'not' a) 'or' c) . z) '&' ((( 'not' c) 'or' a) . z)))) by BVFUNC_1:def 4

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' a) . z))) 'or' (((( 'not' b) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b '&' ( 'not' c)) . z) 'or' ((c '&' ( 'not' b)) . z)) 'or' (((( 'not' a) 'or' c) . z) '&' ((( 'not' c) 'or' a) . z)))) by BVFUNC_1:def 4

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' a) . z))) 'or' (((( 'not' b) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c '&' ( 'not' b)) . z)) 'or' (((( 'not' a) 'or' c) . z) '&' ((( 'not' c) 'or' a) . z)))) by MARGREL1:def 20

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' a) . z))) 'or' (((( 'not' b) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) 'or' (((( 'not' a) 'or' c) . z) '&' ((( 'not' c) 'or' a) . z)))) by MARGREL1:def 20

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' a) . z))) 'or' (((( 'not' b) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) 'or' (((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' c) 'or' a) . z)))) by BVFUNC_1:def 4

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' a) . z))) 'or' (((( 'not' b) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) 'or' (((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (a . z))))) by BVFUNC_1:def 4

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (a . z)))) 'or' (((( 'not' b) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) 'or' (((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (a . z))))) by MARGREL1:def 19

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (a . z)))) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) 'or' (((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (a . z))))) by MARGREL1:def 19

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (a . z)))) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (a . z))))) by MARGREL1:def 19

      .= (((((a . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (a . z)))) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' ((( 'not' (a . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (a . z))))) by MARGREL1:def 19;

      ((b 'imp' a) . z) = TRUE by A2, A1, MARGREL1: 12;

      then

       A8: ( 'not' (b . z)) = TRUE or (a . z) = TRUE by A6, A4;

      now

        per cases by A3, A5;

          case

           A9: ( 'not' (a . z)) = TRUE ;

          then (a . z) = FALSE ;



          then (((a 'eqv' c) 'eqv' (b 'eqv' c)) . z) = ((( FALSE 'or' ((c . z) '&' TRUE )) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( TRUE 'or' (c . z)) '&' ((( 'not' c) . z) 'or' FALSE )))) by A7

          .= ((( FALSE 'or' (c . z)) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( TRUE 'or' (c . z)) '&' ((( 'not' c) . z) 'or' FALSE ))))

          .= (((c . z) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( TRUE 'or' (c . z)) '&' ((( 'not' c) . z) 'or' FALSE ))))

          .= (((c . z) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( TRUE 'or' (c . z)) '&' (( 'not' c) . z))))

          .= (((c . z) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' ( TRUE '&' (( 'not' c) . z))))

          .= (((c . z) 'or' ((( 'not' (b . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( 'not' c) . z)))

          .= ((((c . z) 'or' ((c . z) 'or' ( 'not' (b . z)))) '&' ((c . z) 'or' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( 'not' c) . z))) by XBOOLEAN: 9

          .= (((((c . z) 'or' (c . z)) 'or' ( 'not' (b . z))) '&' ((c . z) 'or' ((( 'not' c) . z) 'or' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( 'not' c) . z))) by BINARITH: 11

          .= ((((c . z) 'or' ( 'not' (b . z))) '&' (((c . z) 'or' (( 'not' c) . z)) 'or' (b . z))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( 'not' c) . z)))

          .= ((((c . z) 'or' ( 'not' (b . z))) '&' (((c . z) 'or' ( 'not' (c . z))) 'or' (b . z))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( 'not' c) . z))) by MARGREL1:def 19

          .= ((((c . z) 'or' ( 'not' (b . z))) '&' ( TRUE 'or' (b . z))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( 'not' c) . z))) by XBOOLEAN: 102

          .= (( TRUE '&' ((c . z) 'or' ( 'not' (b . z)))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( 'not' c) . z)))

          .= (((c . z) 'or' ( 'not' (b . z))) '&' ((((b . z) '&' (( 'not' c) . z)) 'or' ((c . z) '&' ( 'not' (b . z)))) 'or' (( 'not' c) . z)))

          .= (((c . z) 'or' ( 'not' (b . z))) '&' (((b . z) '&' (( 'not' c) . z)) 'or' ((( 'not' c) . z) 'or' ((c . z) '&' ( 'not' (b . z))))))

          .= (((c . z) 'or' ( 'not' (b . z))) '&' (((b . z) '&' (( 'not' c) . z)) 'or' (((( 'not' c) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' ( 'not' (b . z)))))) by XBOOLEAN: 9

          .= (((c . z) 'or' ( 'not' (b . z))) '&' (((b . z) '&' (( 'not' c) . z)) 'or' ((( 'not' (c . z)) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' ( 'not' (b . z)))))) by MARGREL1:def 19

          .= (((c . z) 'or' ( 'not' (b . z))) '&' (((b . z) '&' (( 'not' c) . z)) 'or' ( TRUE '&' ((( 'not' c) . z) 'or' ( 'not' (b . z)))))) by XBOOLEAN: 102

          .= (((c . z) 'or' ( 'not' (b . z))) '&' (((( 'not' c) . z) 'or' ( 'not' (b . z))) 'or' ((b . z) '&' (( 'not' c) . z))))

          .= (((c . z) 'or' ( 'not' (b . z))) '&' ((((( 'not' c) . z) 'or' ( 'not' (b . z))) 'or' (b . z)) '&' (((( 'not' c) . z) 'or' ( 'not' (b . z))) 'or' (( 'not' c) . z)))) by XBOOLEAN: 9

          .= (((c . z) 'or' ( 'not' (b . z))) '&' (((( 'not' c) . z) 'or' (( 'not' (b . z)) 'or' (b . z))) '&' (((( 'not' c) . z) 'or' ( 'not' (b . z))) 'or' (( 'not' c) . z))))

          .= (((c . z) 'or' ( 'not' (b . z))) '&' (((( 'not' c) . z) 'or' TRUE ) '&' (((( 'not' c) . z) 'or' ( 'not' (b . z))) 'or' (( 'not' c) . z)))) by XBOOLEAN: 102

          .= (((c . z) 'or' ( 'not' (b . z))) '&' ( TRUE '&' (((( 'not' c) . z) 'or' ( 'not' (b . z))) 'or' (( 'not' c) . z))))

          .= (((c . z) 'or' ( 'not' (b . z))) '&' ((( 'not' (b . z)) 'or' (( 'not' c) . z)) 'or' (( 'not' c) . z)))

          .= (((c . z) 'or' ( 'not' (b . z))) '&' (( 'not' (b . z)) 'or' ((( 'not' c) . z) 'or' (( 'not' c) . z)))) by BINARITH: 11

          .= ((( 'not' (b . z)) '&' ((c . z) 'or' ( 'not' (b . z)))) 'or' (((c . z) 'or' ( 'not' (b . z))) '&' (( 'not' c) . z))) by XBOOLEAN: 8

          .= (((( 'not' (b . z)) '&' (c . z)) 'or' (( 'not' (b . z)) '&' ( 'not' (b . z)))) 'or' ((( 'not' c) . z) '&' ((c . z) 'or' ( 'not' (b . z))))) by XBOOLEAN: 8

          .= (((( 'not' (b . z)) '&' (c . z)) 'or' ( 'not' (b . z))) 'or' (((( 'not' c) . z) '&' (c . z)) 'or' ((( 'not' c) . z) '&' ( 'not' (b . z))))) by XBOOLEAN: 8

          .= (((( 'not' (b . z)) '&' (c . z)) 'or' ( 'not' (b . z))) 'or' (((c . z) '&' ( 'not' (c . z))) 'or' ((( 'not' c) . z) '&' ( 'not' (b . z))))) by MARGREL1:def 19

          .= (((( 'not' (b . z)) '&' (c . z)) 'or' ( 'not' (b . z))) 'or' ( FALSE 'or' ((( 'not' c) . z) '&' ( 'not' (b . z))))) by XBOOLEAN: 138

          .= ((( 'not' (b . z)) 'or' (( 'not' (b . z)) '&' (c . z))) 'or' ((( 'not' c) . z) '&' ( 'not' (b . z))))

          .= (( 'not' (b . z)) 'or' ((( 'not' (b . z)) '&' (c . z)) 'or' ((( 'not' c) . z) '&' ( 'not' (b . z)))))

          .= TRUE by A8, A9;

          hence thesis;

        end;

          case (b . z) = TRUE ;

          then ( 'not' (b . z)) = FALSE ;



          then (((a 'eqv' c) 'eqv' (b 'eqv' c)) . z) = ((((( 'not' c) . z) 'or' ( FALSE '&' (c . z))) 'or' (( FALSE 'or' (c . z)) '&' ((( 'not' c) . z) 'or' TRUE ))) '&' (((( 'not' c) . z) 'or' ((c . z) '&' FALSE )) 'or' (( FALSE 'or' (c . z)) '&' ((( 'not' c) . z) 'or' TRUE )))) by A2, A1, A6, A4, A7

          .= ((((( 'not' c) . z) 'or' FALSE ) 'or' (( FALSE 'or' (c . z)) '&' ((( 'not' c) . z) 'or' TRUE ))) '&' (((( 'not' c) . z) 'or' FALSE ) 'or' (( FALSE 'or' (c . z)) '&' ((( 'not' c) . z) 'or' TRUE ))))

          .= ((((( 'not' c) . z) 'or' FALSE ) 'or' (( FALSE 'or' (c . z)) '&' TRUE )) '&' (((( 'not' c) . z) 'or' FALSE ) 'or' (( FALSE 'or' (c . z)) '&' TRUE )))

          .= ((((( 'not' c) . z) 'or' FALSE ) 'or' ((c . z) '&' TRUE )) '&' (((( 'not' c) . z) 'or' FALSE ) 'or' ((c . z) '&' TRUE )))

          .= (((( 'not' c) . z) 'or' ( TRUE '&' (c . z))) '&' ((( 'not' c) . z) 'or' ((c . z) '&' TRUE )))

          .= (((( 'not' c) . z) 'or' (c . z)) '&' ((( 'not' c) . z) 'or' (c . z)))

          .= ((( 'not' (c . z)) 'or' (c . z)) '&' (( 'not' (c . z)) 'or' (c . z))) by MARGREL1:def 19

          .= TRUE by XBOOLEAN: 102;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:66

    for a,b,c be Function of Y, BOOLEAN holds (a 'eqv' b) '<' ((a 'imp' c) 'eqv' (b 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: ((a 'eqv' b) . z) = (((a 'imp' b) '&' (b 'imp' a)) . z) by BVFUNC_4: 7

      .= (((a 'imp' b) . z) '&' ((b 'imp' a) . z)) by MARGREL1:def 20;

      assume

       A2: ((a 'eqv' b) . z) = TRUE ;

      then ((a 'imp' b) . z) = TRUE by A1, MARGREL1: 12;

      then

       A3: (( 'not' (a . z)) 'or' (b . z)) = TRUE by BVFUNC_1:def 8;

      ((b 'imp' a) . z) = TRUE by A2, A1, MARGREL1: 12;

      then

       A4: (( 'not' (b . z)) 'or' (a . z)) = TRUE by BVFUNC_1:def 8;



       A5: (((a 'imp' c) 'eqv' (b 'imp' c)) . z) = ((((a 'imp' c) 'imp' (b 'imp' c)) '&' ((b 'imp' c) 'imp' (a 'imp' c))) . z) by BVFUNC_4: 7

      .= ((((a 'imp' c) 'imp' (b 'imp' c)) . z) '&' (((b 'imp' c) 'imp' (a 'imp' c)) . z)) by MARGREL1:def 20

      .= ((( 'not' ((a 'imp' c) . z)) 'or' ((b 'imp' c) . z)) '&' (((b 'imp' c) 'imp' (a 'imp' c)) . z)) by BVFUNC_1:def 8

      .= ((( 'not' ((a 'imp' c) . z)) 'or' ((b 'imp' c) . z)) '&' (( 'not' ((b 'imp' c) . z)) 'or' ((a 'imp' c) . z))) by BVFUNC_1:def 8

      .= ((( 'not' (( 'not' (a . z)) 'or' (c . z))) 'or' ((b 'imp' c) . z)) '&' (( 'not' ((b 'imp' c) . z)) 'or' ((a 'imp' c) . z))) by BVFUNC_1:def 8

      .= ((( 'not' (( 'not' (a . z)) 'or' (c . z))) 'or' (( 'not' (b . z)) 'or' (c . z))) '&' (( 'not' ((b 'imp' c) . z)) 'or' ((a 'imp' c) . z))) by BVFUNC_1:def 8

      .= ((( 'not' (( 'not' (a . z)) 'or' (c . z))) 'or' (( 'not' (b . z)) 'or' (c . z))) '&' (( 'not' (( 'not' (b . z)) 'or' (c . z))) 'or' ((a 'imp' c) . z))) by BVFUNC_1:def 8

      .= ((((a . z) '&' ( 'not' (c . z))) 'or' (( 'not' (b . z)) 'or' (c . z))) '&' (((b . z) '&' ( 'not' (c . z))) 'or' (( 'not' (a . z)) 'or' (c . z)))) by BVFUNC_1:def 8;



       A6: (a . z) = TRUE or (a . z) = FALSE by XBOOLEAN:def 3;



       A7: ((b 'imp' a) . z) = (( 'not' (b . z)) 'or' (a . z)) by BVFUNC_1:def 8;



       A8: (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;

      now

        per cases by A3, A8;

          case

           A9: ( 'not' (a . z)) = TRUE ;

          then (a . z) = FALSE ;

          then ( 'not' (b . z)) = TRUE by A4;



          then (((a 'imp' c) 'eqv' (b 'imp' c)) . z) = (( FALSE 'or' ( TRUE 'or' (c . z))) '&' ( FALSE 'or' ( TRUE 'or' (c . z)))) by A5, A9

          .= (( TRUE 'or' (c . z)) '&' ( TRUE 'or' (c . z)))

          .= TRUE ;

          hence thesis;

        end;

          case (b . z) = TRUE ;

          then ( 'not' (b . z)) = FALSE ;



          then (((a 'imp' c) 'eqv' (b 'imp' c)) . z) = ((( 'not' (c . z)) 'or' ( FALSE 'or' (c . z))) '&' (( 'not' (c . z)) 'or' ( FALSE 'or' (c . z)))) by A2, A1, A7, A6, A5

          .= ((( 'not' (c . z)) 'or' (c . z)) '&' (( 'not' (c . z)) 'or' (c . z)))

          .= TRUE by XBOOLEAN: 102;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:67

    for a,b,c be Function of Y, BOOLEAN holds (a 'eqv' b) '<' ((c 'imp' a) 'eqv' (c 'imp' b))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: ((a 'eqv' b) . z) = (((a 'imp' b) '&' (b 'imp' a)) . z) by BVFUNC_4: 7

      .= (((a 'imp' b) . z) '&' ((b 'imp' a) . z)) by MARGREL1:def 20;

      assume

       A2: ((a 'eqv' b) . z) = TRUE ;

      then ((a 'imp' b) . z) = TRUE by A1, MARGREL1: 12;

      then

       A3: (( 'not' (a . z)) 'or' (b . z)) = TRUE by BVFUNC_1:def 8;

      ((b 'imp' a) . z) = TRUE by A2, A1, MARGREL1: 12;

      then

       A4: (( 'not' (b . z)) 'or' (a . z)) = TRUE by BVFUNC_1:def 8;



       A5: (((c 'imp' a) 'eqv' (c 'imp' b)) . z) = ((((c 'imp' a) 'imp' (c 'imp' b)) '&' ((c 'imp' b) 'imp' (c 'imp' a))) . z) by BVFUNC_4: 7

      .= ((((c 'imp' a) 'imp' (c 'imp' b)) . z) '&' (((c 'imp' b) 'imp' (c 'imp' a)) . z)) by MARGREL1:def 20

      .= ((( 'not' ((c 'imp' a) . z)) 'or' ((c 'imp' b) . z)) '&' (((c 'imp' b) 'imp' (c 'imp' a)) . z)) by BVFUNC_1:def 8

      .= ((( 'not' ((c 'imp' a) . z)) 'or' ((c 'imp' b) . z)) '&' (( 'not' ((c 'imp' b) . z)) 'or' ((c 'imp' a) . z))) by BVFUNC_1:def 8

      .= ((( 'not' (( 'not' (c . z)) 'or' (a . z))) 'or' ((c 'imp' b) . z)) '&' (( 'not' ((c 'imp' b) . z)) 'or' ((c 'imp' a) . z))) by BVFUNC_1:def 8

      .= ((( 'not' (( 'not' (c . z)) 'or' (a . z))) 'or' (( 'not' (c . z)) 'or' (b . z))) '&' (( 'not' ((c 'imp' b) . z)) 'or' ((c 'imp' a) . z))) by BVFUNC_1:def 8

      .= ((( 'not' (( 'not' (c . z)) 'or' (a . z))) 'or' (( 'not' (c . z)) 'or' (b . z))) '&' (( 'not' (( 'not' (c . z)) 'or' (b . z))) 'or' ((c 'imp' a) . z))) by BVFUNC_1:def 8

      .= ((((c . z) '&' ( 'not' (a . z))) 'or' (( 'not' (c . z)) 'or' (b . z))) '&' (((c . z) '&' ( 'not' (b . z))) 'or' (( 'not' (c . z)) 'or' (a . z)))) by BVFUNC_1:def 8;



       A7: ((b 'imp' a) . z) = (( 'not' (b . z)) 'or' (a . z)) by BVFUNC_1:def 8;



       A8: (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;

      now

        per cases by A3, A8;

          case

           A9: ( 'not' (a . z)) = TRUE ;

          then (a . z) = FALSE ;

          then ( 'not' (b . z)) = TRUE by A4;



          then (((c 'imp' a) 'eqv' (c 'imp' b)) . z) = (((c . z) 'or' (( 'not' (c . z)) 'or' FALSE )) '&' ((c . z) 'or' (( 'not' (c . z)) 'or' FALSE ))) by A5, A9

          .= (((c . z) 'or' ( 'not' (c . z))) '&' ((c . z) 'or' ( 'not' (c . z))))

          .= TRUE by XBOOLEAN: 102;

          hence thesis;

        end;

          case (b . z) = TRUE ;

          then ( 'not' (b . z)) = FALSE ;



          then (((c 'imp' a) 'eqv' (c 'imp' b)) . z) = (( FALSE 'or' (( 'not' (c . z)) 'or' TRUE )) '&' ( FALSE 'or' (( 'not' (c . z)) 'or' TRUE ))) by A2, A1, A7, A5, MARGREL1: 12

          .= ((( 'not' (c . z)) 'or' TRUE ) '&' (( 'not' (c . z)) 'or' TRUE ))

          .= TRUE ;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:68

    for a,b,c be Function of Y, BOOLEAN holds (a 'eqv' b) '<' ((a '&' c) 'eqv' (b '&' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: ((a 'eqv' b) . z) = (((a 'imp' b) '&' (b 'imp' a)) . z) by BVFUNC_4: 7

      .= (((a 'imp' b) . z) '&' ((b 'imp' a) . z)) by MARGREL1:def 20;

      assume

       A2: ((a 'eqv' b) . z) = TRUE ;

      then ((a 'imp' b) . z) = TRUE by A1, MARGREL1: 12;

      then

       A3: (( 'not' (a . z)) 'or' (b . z)) = TRUE by BVFUNC_1:def 8;

      ((b 'imp' a) . z) = TRUE by A2, A1, MARGREL1: 12;

      then

       A4: (( 'not' (b . z)) 'or' (a . z)) = TRUE by BVFUNC_1:def 8;



       A5: (((a '&' c) 'eqv' (b '&' c)) . z) = ((((a '&' c) 'imp' (b '&' c)) '&' ((b '&' c) 'imp' (a '&' c))) . z) by BVFUNC_4: 7

      .= ((((a '&' c) 'imp' (b '&' c)) . z) '&' (((b '&' c) 'imp' (a '&' c)) . z)) by MARGREL1:def 20

      .= ((( 'not' ((a '&' c) . z)) 'or' ((b '&' c) . z)) '&' (((b '&' c) 'imp' (a '&' c)) . z)) by BVFUNC_1:def 8

      .= ((( 'not' ((a '&' c) . z)) 'or' ((b '&' c) . z)) '&' (( 'not' ((b '&' c) . z)) 'or' ((a '&' c) . z))) by BVFUNC_1:def 8

      .= ((( 'not' ((a . z) '&' (c . z))) 'or' ((b '&' c) . z)) '&' (( 'not' ((b '&' c) . z)) 'or' ((a '&' c) . z))) by MARGREL1:def 20

      .= ((( 'not' ((a . z) '&' (c . z))) 'or' ((b . z) '&' (c . z))) '&' (( 'not' ((b '&' c) . z)) 'or' ((a '&' c) . z))) by MARGREL1:def 20

      .= ((( 'not' ((a . z) '&' (c . z))) 'or' ((b . z) '&' (c . z))) '&' (( 'not' ((b . z) '&' (c . z))) 'or' ((a '&' c) . z))) by MARGREL1:def 20

      .= (((( 'not' (a . z)) 'or' ( 'not' (c . z))) 'or' ((b . z) '&' (c . z))) '&' ((( 'not' (b . z)) 'or' ( 'not' (c . z))) 'or' ((a . z) '&' (c . z)))) by MARGREL1:def 20;



       A6: (a . z) = TRUE or (a . z) = FALSE by XBOOLEAN:def 3;



       A7: ((b 'imp' a) . z) = (( 'not' (b . z)) 'or' (a . z)) by BVFUNC_1:def 8;



       A8: (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;

      now

        per cases by A3, A8;

          case

           A9: ( 'not' (a . z)) = TRUE ;

          then (a . z) = FALSE ;

          then ( 'not' (b . z)) = TRUE by A4;



          then (((a '&' c) 'eqv' (b '&' c)) . z) = ((( TRUE 'or' ( 'not' (c . z))) 'or' FALSE ) '&' (( TRUE 'or' ( 'not' (c . z))) 'or' FALSE )) by A5, A9

          .= (( TRUE 'or' ( 'not' (c . z))) '&' ( TRUE 'or' ( 'not' (c . z))))

          .= TRUE ;

          hence thesis;

        end;

          case (b . z) = TRUE ;

          then ( 'not' (b . z)) = FALSE ;



          then (((a '&' c) 'eqv' (b '&' c)) . z) = ((( FALSE 'or' ( 'not' (c . z))) 'or' (c . z)) '&' (( FALSE 'or' ( 'not' (c . z))) 'or' (c . z))) by A2, A1, A7, A6, A5

          .= ((( 'not' (c . z)) 'or' (c . z)) '&' (( 'not' (c . z)) 'or' (c . z)))

          .= TRUE by XBOOLEAN: 102;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:69

    for a,b,c be Function of Y, BOOLEAN holds (a 'eqv' b) '<' ((a 'or' c) 'eqv' (b 'or' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: ((a 'eqv' b) . z) = (((a 'imp' b) '&' (b 'imp' a)) . z) by BVFUNC_4: 7

      .= (((a 'imp' b) . z) '&' ((b 'imp' a) . z)) by MARGREL1:def 20;

      assume

       A2: ((a 'eqv' b) . z) = TRUE ;

      then ((a 'imp' b) . z) = TRUE by A1, MARGREL1: 12;

      then

       A3: (( 'not' (a . z)) 'or' (b . z)) = TRUE by BVFUNC_1:def 8;

      ((b 'imp' a) . z) = TRUE by A2, A1, MARGREL1: 12;

      then

       A4: (( 'not' (b . z)) 'or' (a . z)) = TRUE by BVFUNC_1:def 8;



       A5: (((a 'or' c) 'eqv' (b 'or' c)) . z) = ((((a 'or' c) 'imp' (b 'or' c)) '&' ((b 'or' c) 'imp' (a 'or' c))) . z) by BVFUNC_4: 7

      .= ((((a 'or' c) 'imp' (b 'or' c)) . z) '&' (((b 'or' c) 'imp' (a 'or' c)) . z)) by MARGREL1:def 20

      .= ((( 'not' ((a 'or' c) . z)) 'or' ((b 'or' c) . z)) '&' (((b 'or' c) 'imp' (a 'or' c)) . z)) by BVFUNC_1:def 8

      .= ((( 'not' ((a 'or' c) . z)) 'or' ((b 'or' c) . z)) '&' (( 'not' ((b 'or' c) . z)) 'or' ((a 'or' c) . z))) by BVFUNC_1:def 8

      .= ((( 'not' ((a . z) 'or' (c . z))) 'or' ((b 'or' c) . z)) '&' (( 'not' ((b 'or' c) . z)) 'or' ((a 'or' c) . z))) by BVFUNC_1:def 4

      .= ((( 'not' ((a . z) 'or' (c . z))) 'or' ((b . z) 'or' (c . z))) '&' (( 'not' ((b 'or' c) . z)) 'or' ((a 'or' c) . z))) by BVFUNC_1:def 4

      .= ((( 'not' ((a . z) 'or' (c . z))) 'or' ((b . z) 'or' (c . z))) '&' (( 'not' ((b . z) 'or' (c . z))) 'or' ((a 'or' c) . z))) by BVFUNC_1:def 4

      .= (((( 'not' (a . z)) '&' ( 'not' (c . z))) 'or' ((b . z) 'or' (c . z))) '&' ((( 'not' (b . z)) '&' ( 'not' (c . z))) 'or' ((a . z) 'or' (c . z)))) by BVFUNC_1:def 4;



       A6: (a . z) = TRUE or (a . z) = FALSE by XBOOLEAN:def 3;



       A7: ((b 'imp' a) . z) = (( 'not' (b . z)) 'or' (a . z)) by BVFUNC_1:def 8;



       A8: (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;

      now

        per cases by A3, A8;

          case

           A9: ( 'not' (a . z)) = TRUE ;

          then (a . z) = FALSE ;

          then ( 'not' (b . z)) = TRUE by A4;



          then (((a 'or' c) 'eqv' (b 'or' c)) . z) = ((( 'not' (c . z)) 'or' ( FALSE 'or' (c . z))) '&' (( 'not' (c . z)) 'or' ( FALSE 'or' (c . z)))) by A5, A9

          .= ((( 'not' (c . z)) 'or' (c . z)) '&' (( 'not' (c . z)) 'or' (c . z)))

          .= TRUE by XBOOLEAN: 102;

          hence thesis;

        end;

          case (b . z) = TRUE ;

          then ( 'not' (b . z)) = FALSE ;



          then (((a 'or' c) 'eqv' (b 'or' c)) . z) = (( FALSE 'or' ( TRUE 'or' (c . z))) '&' ( FALSE 'or' ( TRUE 'or' (c . z)))) by A2, A1, A7, A6, A5

          .= (( TRUE 'or' (c . z)) '&' ( TRUE 'or' (c . z)))

          .= TRUE ;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:70

    for a,b be Function of Y, BOOLEAN holds a '<' (((a 'eqv' b) 'eqv' (b 'eqv' a)) 'eqv' a)

    proof

      let a,b be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (( 'not' a) . z) = ( 'not' (a . z)) by MARGREL1:def 19;

      assume

       A2: (a . z) = TRUE ;

      then

       A3: ( 'not' (a . z)) = FALSE ;



       A4: (((a 'eqv' b) 'eqv' (b 'eqv' a)) . z) = ((((a 'eqv' b) 'imp' (b 'eqv' a)) '&' ((b 'eqv' a) 'imp' (a 'eqv' b))) . z) by BVFUNC_4: 7

      .= (((( 'not' (a 'eqv' b)) 'or' (b 'eqv' a)) . z) '&' ((( 'not' (b 'eqv' a)) 'or' (a 'eqv' b)) . z)) by BVFUNC_4: 8

      .= (((( 'not' (a 'eqv' b)) . z) 'or' ((b 'eqv' a) . z)) '&' ((( 'not' (b 'eqv' a)) . z) 'or' ((a 'eqv' b) . z))) by BVFUNC_1:def 4

      .= (((( 'not' ((a 'imp' b) '&' (b 'imp' a))) . z) 'or' ((b 'eqv' a) . z)) '&' ((( 'not' (b 'eqv' a)) . z) 'or' ((a 'eqv' b) . z))) by BVFUNC_4: 7

      .= (((( 'not' ((a 'imp' b) '&' (b 'imp' a))) . z) 'or' ((b 'eqv' a) . z)) '&' ((( 'not' (b 'eqv' a)) . z) 'or' (((a 'imp' b) '&' (b 'imp' a)) . z))) by BVFUNC_4: 7

      .= (((( 'not' ((a 'imp' b) '&' (b 'imp' a))) . z) 'or' (((b 'imp' a) '&' (a 'imp' b)) . z)) '&' ((( 'not' (b 'eqv' a)) . z) 'or' (((a 'imp' b) '&' (b 'imp' a)) . z))) by BVFUNC_4: 7

      .= (((( 'not' ((a 'imp' b) '&' (b 'imp' a))) . z) 'or' (((b 'imp' a) '&' (a 'imp' b)) . z)) '&' ((( 'not' ((b 'imp' a) '&' (a 'imp' b))) . z) 'or' (((a 'imp' b) '&' (b 'imp' a)) . z))) by BVFUNC_4: 7

      .= ((( 'not' (((a 'imp' b) '&' (b 'imp' a)) . z)) 'or' (((b 'imp' a) '&' (a 'imp' b)) . z)) '&' (( 'not' (((b 'imp' a) '&' (a 'imp' b)) . z)) 'or' (((a 'imp' b) '&' (b 'imp' a)) . z))) by MARGREL1:def 19

      .= ((( 'not' (((a 'imp' b) . z) '&' ((b 'imp' a) . z))) 'or' (((b 'imp' a) '&' (a 'imp' b)) . z)) '&' (( 'not' (((b 'imp' a) '&' (a 'imp' b)) . z)) 'or' (((a 'imp' b) '&' (b 'imp' a)) . z))) by MARGREL1:def 20

      .= ((( 'not' (((a 'imp' b) . z) '&' ((b 'imp' a) . z))) 'or' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) '&' (( 'not' (((b 'imp' a) '&' (a 'imp' b)) . z)) 'or' (((a 'imp' b) '&' (b 'imp' a)) . z))) by MARGREL1:def 20

      .= ((( 'not' (((a 'imp' b) . z) '&' ((b 'imp' a) . z))) 'or' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) '&' (( 'not' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) 'or' (((a 'imp' b) '&' (b 'imp' a)) . z))) by MARGREL1:def 20

      .= ((( 'not' (((a 'imp' b) . z) '&' ((b 'imp' a) . z))) 'or' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) '&' (( 'not' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) 'or' (((a 'imp' b) . z) '&' ((b 'imp' a) . z)))) by MARGREL1:def 20

      .= ((( 'not' (((( 'not' a) 'or' b) . z) '&' ((b 'imp' a) . z))) 'or' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) '&' (( 'not' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) 'or' (((a 'imp' b) . z) '&' ((b 'imp' a) . z)))) by BVFUNC_4: 8

      .= ((( 'not' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z))) 'or' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) '&' (( 'not' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) 'or' (((a 'imp' b) . z) '&' ((b 'imp' a) . z)))) by BVFUNC_4: 8

      .= ((( 'not' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z))) 'or' (((( 'not' b) 'or' a) . z) '&' ((a 'imp' b) . z))) '&' (( 'not' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) 'or' (((a 'imp' b) . z) '&' ((b 'imp' a) . z)))) by BVFUNC_4: 8

      .= ((( 'not' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z))) 'or' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) '&' (( 'not' (((b 'imp' a) . z) '&' ((a 'imp' b) . z))) 'or' (((a 'imp' b) . z) '&' ((b 'imp' a) . z)))) by BVFUNC_4: 8

      .= ((( 'not' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z))) 'or' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) '&' (( 'not' (((( 'not' b) 'or' a) . z) '&' ((a 'imp' b) . z))) 'or' (((a 'imp' b) . z) '&' ((b 'imp' a) . z)))) by BVFUNC_4: 8

      .= ((( 'not' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z))) 'or' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) '&' (( 'not' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) 'or' (((a 'imp' b) . z) '&' ((b 'imp' a) . z)))) by BVFUNC_4: 8

      .= ((( 'not' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z))) 'or' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) '&' (( 'not' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) 'or' (((( 'not' a) 'or' b) . z) '&' ((b 'imp' a) . z)))) by BVFUNC_4: 8

      .= ((( 'not' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z))) 'or' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) '&' (( 'not' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) 'or' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z)))) by BVFUNC_4: 8

      .= ((( 'not' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' a) . z))) 'or' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) '&' (( 'not' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) 'or' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z)))) by BVFUNC_1:def 4

      .= ((( 'not' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (a . z)))) 'or' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) '&' (( 'not' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) 'or' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z)))) by BVFUNC_1:def 4

      .= ((( 'not' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (a . z)))) 'or' (((( 'not' b) . z) 'or' (a . z)) '&' ((( 'not' a) 'or' b) . z))) '&' (( 'not' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) 'or' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z)))) by BVFUNC_1:def 4

      .= ((( 'not' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (a . z)))) 'or' (((( 'not' b) . z) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (b . z)))) '&' (( 'not' (((( 'not' b) 'or' a) . z) '&' ((( 'not' a) 'or' b) . z))) 'or' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z)))) by BVFUNC_1:def 4

      .= ((( 'not' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (a . z)))) 'or' (((( 'not' b) . z) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (b . z)))) '&' (( 'not' (((( 'not' b) . z) 'or' (a . z)) '&' ((( 'not' a) 'or' b) . z))) 'or' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z)))) by BVFUNC_1:def 4

      .= ((( 'not' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (a . z)))) 'or' (((( 'not' b) . z) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (b . z)))) '&' (( 'not' (((( 'not' b) . z) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (b . z)))) 'or' (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' a) . z)))) by BVFUNC_1:def 4

      .= ((( 'not' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (a . z)))) 'or' (((( 'not' b) . z) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (b . z)))) '&' (( 'not' (((( 'not' b) . z) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (b . z)))) 'or' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' a) . z)))) by BVFUNC_1:def 4

      .= ((( 'not' (( FALSE 'or' (b . z)) '&' ((( 'not' b) . z) 'or' TRUE ))) 'or' (((( 'not' b) . z) 'or' TRUE ) '&' ( FALSE 'or' (b . z)))) '&' (( 'not' (((( 'not' b) . z) 'or' TRUE ) '&' ( FALSE 'or' (b . z)))) 'or' (( FALSE 'or' (b . z)) '&' ((( 'not' b) . z) 'or' TRUE )))) by A3, A1, BVFUNC_1:def 4

      .= ((( 'not' ((b . z) '&' ((( 'not' b) . z) 'or' TRUE ))) 'or' (((( 'not' b) . z) 'or' TRUE ) '&' ( FALSE 'or' (b . z)))) '&' (( 'not' (((( 'not' b) . z) 'or' TRUE ) '&' ( FALSE 'or' (b . z)))) 'or' (( FALSE 'or' (b . z)) '&' ((( 'not' b) . z) 'or' TRUE ))))

      .= ((( 'not' ((b . z) '&' ((( 'not' b) . z) 'or' TRUE ))) 'or' (((( 'not' b) . z) 'or' TRUE ) '&' (b . z))) '&' (( 'not' (((( 'not' b) . z) 'or' TRUE ) '&' ( FALSE 'or' (b . z)))) 'or' (( FALSE 'or' (b . z)) '&' ((( 'not' b) . z) 'or' TRUE ))))

      .= ((( 'not' ((b . z) '&' ((( 'not' b) . z) 'or' TRUE ))) 'or' (((( 'not' b) . z) 'or' TRUE ) '&' (b . z))) '&' (( 'not' (((( 'not' b) . z) 'or' TRUE ) '&' ( FALSE 'or' (b . z)))) 'or' ((b . z) '&' ((( 'not' b) . z) 'or' TRUE ))))

      .= ((( 'not' ((b . z) '&' TRUE )) 'or' (((( 'not' b) . z) 'or' TRUE ) '&' (b . z))) '&' (( 'not' (((( 'not' b) . z) 'or' TRUE ) '&' ( FALSE 'or' (b . z)))) 'or' ((b . z) '&' ((( 'not' b) . z) 'or' TRUE ))))

      .= ((( 'not' ((b . z) '&' TRUE )) 'or' ( TRUE '&' (b . z))) '&' (( 'not' (((( 'not' b) . z) 'or' TRUE ) '&' ( FALSE 'or' (b . z)))) 'or' ((b . z) '&' ((( 'not' b) . z) 'or' TRUE ))))

      .= ((( 'not' ((b . z) '&' TRUE )) 'or' ( TRUE '&' (b . z))) '&' (( 'not' ( TRUE '&' ( FALSE 'or' (b . z)))) 'or' ((b . z) '&' ((( 'not' b) . z) 'or' TRUE ))))

      .= ((( 'not' ((b . z) '&' TRUE )) 'or' ( TRUE '&' (b . z))) '&' (( 'not' ( TRUE '&' ( FALSE 'or' (b . z)))) 'or' ((b . z) '&' TRUE )))

      .= ((( 'not' ( TRUE '&' (b . z))) 'or' ( TRUE '&' (b . z))) '&' (( 'not' ( TRUE '&' (b . z))) 'or' ((b . z) '&' TRUE )))

      .= ((( 'not' (b . z)) 'or' ( TRUE '&' (b . z))) '&' (( 'not' (b . z)) 'or' ( TRUE '&' (b . z))))

      .= ((( 'not' (b . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' (b . z)))

      .= TRUE by XBOOLEAN: 102;

      ((((a 'eqv' b) 'eqv' (b 'eqv' a)) 'eqv' a) . z) = (((((a 'eqv' b) 'eqv' (b 'eqv' a)) 'imp' a) '&' (a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a)))) . z) by BVFUNC_4: 7

      .= (((((a 'eqv' b) 'eqv' (b 'eqv' a)) 'imp' a) . z) '&' ((a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a))) . z)) by MARGREL1:def 20

      .= ((( 'not' (((a 'eqv' b) 'eqv' (b 'eqv' a)) . z)) 'or' (a . z)) '&' ((a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a))) . z)) by BVFUNC_1:def 8

      .= ((( 'not' (((a 'eqv' b) 'eqv' (b 'eqv' a)) . z)) 'or' (a . z)) '&' (( 'not' (a . z)) 'or' (((a 'eqv' b) 'eqv' (b 'eqv' a)) . z))) by BVFUNC_1:def 8

      .= (( FALSE 'or' (a . z)) '&' (( 'not' (a . z)) 'or' TRUE )) by A4

      .= ((a . z) '&' (( 'not' (a . z)) 'or' TRUE ))

      .= ( TRUE '&' (a . z))

      .= TRUE by A2;

      hence thesis;

    end;

    theorem :: BVFUNC_6:71

    for a,b be Function of Y, BOOLEAN holds a '<' ((a 'imp' b) 'eqv' b)

    proof

      let a,b be Function of Y, BOOLEAN ;

      let z be Element of Y;

      assume (a . z) = TRUE ;

      then

       A2: ( 'not' (a . z)) = FALSE ;

      (((a 'imp' b) 'eqv' b) . z) = (((( 'not' a) 'or' b) 'eqv' b) . z) by BVFUNC_4: 8

      .= ((((( 'not' a) 'or' b) 'imp' b) '&' (b 'imp' (( 'not' a) 'or' b))) . z) by BVFUNC_4: 7

      .= (((( 'not' (( 'not' a) 'or' b)) 'or' b) '&' (b 'imp' (( 'not' a) 'or' b))) . z) by BVFUNC_4: 8

      .= (((( 'not' (( 'not' a) 'or' b)) 'or' b) '&' (( 'not' b) 'or' (( 'not' a) 'or' b))) . z) by BVFUNC_4: 8

      .= (((( 'not' (( 'not' a) 'or' b)) 'or' b) . z) '&' ((( 'not' b) 'or' (( 'not' a) 'or' b)) . z)) by MARGREL1:def 20

      .= (((( 'not' (( 'not' a) 'or' b)) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' (( 'not' a) 'or' b)) . z)) by BVFUNC_1:def 4

      .= ((( 'not' ((( 'not' a) 'or' b) . z)) 'or' (b . z)) '&' ((( 'not' b) 'or' (( 'not' a) 'or' b)) . z)) by MARGREL1:def 19

      .= ((( 'not' ((( 'not' a) . z) 'or' (b . z))) 'or' (b . z)) '&' ((( 'not' b) 'or' (( 'not' a) 'or' b)) . z)) by BVFUNC_1:def 4

      .= (((( 'not' ( 'not' (a . z))) '&' ( 'not' (b . z))) 'or' (b . z)) '&' ((( 'not' b) 'or' (( 'not' a) 'or' b)) . z)) by MARGREL1:def 19

      .= ((((a . z) '&' ( 'not' (b . z))) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' ((( 'not' a) 'or' b) . z))) by BVFUNC_1:def 4

      .= ((((a . z) '&' ( 'not' (b . z))) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' ((( 'not' a) . z) 'or' (b . z)))) by BVFUNC_1:def 4

      .= ((((a . z) '&' ( 'not' (b . z))) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (( 'not' (a . z)) 'or' (b . z)))) by MARGREL1:def 19

      .= ((( TRUE '&' ( 'not' (b . z))) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' ( FALSE 'or' (b . z)))) by A2, MARGREL1:def 19

      .= ((( 'not' (b . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' ( FALSE 'or' (b . z))))

      .= ((( 'not' (b . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' (b . z)))

      .= TRUE by XBOOLEAN: 102;

      hence thesis;

    end;

    theorem :: BVFUNC_6:72

    for a,b be Function of Y, BOOLEAN holds a '<' ((b 'imp' a) 'eqv' a)

    proof

      let a,b be Function of Y, BOOLEAN ;

      let z be Element of Y;

      assume (a . z) = TRUE ;

      then

       A2: ( 'not' (a . z)) = FALSE ;

      (((b 'imp' a) 'eqv' a) . z) = (((( 'not' b) 'or' a) 'eqv' a) . z) by BVFUNC_4: 8

      .= ((((( 'not' b) 'or' a) 'imp' a) '&' (a 'imp' (( 'not' b) 'or' a))) . z) by BVFUNC_4: 7

      .= (((( 'not' (( 'not' b) 'or' a)) 'or' a) '&' (a 'imp' (( 'not' b) 'or' a))) . z) by BVFUNC_4: 8

      .= (((( 'not' (( 'not' b) 'or' a)) 'or' a) '&' (( 'not' a) 'or' (( 'not' b) 'or' a))) . z) by BVFUNC_4: 8

      .= (((( 'not' (( 'not' b) 'or' a)) 'or' a) . z) '&' ((( 'not' a) 'or' (( 'not' b) 'or' a)) . z)) by MARGREL1:def 20

      .= (((( 'not' (( 'not' b) 'or' a)) . z) 'or' (a . z)) '&' ((( 'not' a) 'or' (( 'not' b) 'or' a)) . z)) by BVFUNC_1:def 4

      .= ((( 'not' ((( 'not' b) 'or' a) . z)) 'or' (a . z)) '&' ((( 'not' a) 'or' (( 'not' b) 'or' a)) . z)) by MARGREL1:def 19

      .= ((( 'not' ((( 'not' b) . z) 'or' (a . z))) 'or' (a . z)) '&' ((( 'not' a) 'or' (( 'not' b) 'or' a)) . z)) by BVFUNC_1:def 4

      .= (((( 'not' ( 'not' (b . z))) '&' ( 'not' (a . z))) 'or' (a . z)) '&' ((( 'not' a) 'or' (( 'not' b) 'or' a)) . z)) by MARGREL1:def 19

      .= ((((b . z) '&' ( 'not' (a . z))) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' ((( 'not' b) 'or' a) . z))) by BVFUNC_1:def 4

      .= ((((b . z) '&' ( 'not' (a . z))) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' ((( 'not' b) . z) 'or' (a . z)))) by BVFUNC_1:def 4

      .= ((((b . z) '&' ( 'not' (a . z))) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (( 'not' (b . z)) 'or' (a . z)))) by MARGREL1:def 19

      .= ((((b . z) '&' ( 'not' (a . z))) 'or' (a . z)) '&' (( 'not' (a . z)) 'or' (( 'not' (b . z)) 'or' (a . z)))) by MARGREL1:def 19

      .= ( TRUE '&' ( FALSE 'or' (( 'not' (b . z)) 'or' TRUE ))) by A2

      .= ( FALSE 'or' (( 'not' (b . z)) 'or' TRUE ))

      .= (( 'not' (b . z)) 'or' TRUE )

      .= TRUE ;

      hence thesis;

    end;

    theorem :: BVFUNC_6:73

    for a,b be Function of Y, BOOLEAN holds a '<' (((a '&' b) 'eqv' (b '&' a)) 'eqv' a)

    proof

      let a,b be Function of Y, BOOLEAN ;

      let z be Element of Y;

      assume

       A1: (a . z) = TRUE ;



       A2: (((a '&' b) 'eqv' (a '&' b)) . z) = ((((a '&' b) 'imp' (a '&' b)) '&' ((a '&' b) 'imp' (a '&' b))) . z) by BVFUNC_4: 7

      .= ((( 'not' (a '&' b)) 'or' (a '&' b)) . z) by BVFUNC_4: 8

      .= (( I_el Y) . z) by BVFUNC_4: 6

      .= TRUE by BVFUNC_1:def 11;

      ((((a '&' b) 'eqv' (b '&' a)) 'eqv' a) . z) = (((((a '&' b) 'eqv' (a '&' b)) 'imp' a) '&' (a 'imp' ((a '&' b) 'eqv' (a '&' b)))) . z) by BVFUNC_4: 7

      .= (((((a '&' b) 'eqv' (a '&' b)) 'imp' a) . z) '&' ((a 'imp' ((a '&' b) 'eqv' (a '&' b))) . z)) by MARGREL1:def 20

      .= (((( 'not' ((a '&' b) 'eqv' (a '&' b))) 'or' a) . z) '&' ((a 'imp' ((a '&' b) 'eqv' (a '&' b))) . z)) by BVFUNC_4: 8

      .= (((( 'not' ((a '&' b) 'eqv' (a '&' b))) 'or' a) . z) '&' ((( 'not' a) 'or' ((a '&' b) 'eqv' (a '&' b))) . z)) by BVFUNC_4: 8

      .= (((( 'not' ((a '&' b) 'eqv' (a '&' b))) . z) 'or' (a . z)) '&' ((( 'not' a) 'or' ((a '&' b) 'eqv' (a '&' b))) . z)) by BVFUNC_1:def 4

      .= (((( 'not' ((a '&' b) 'eqv' (a '&' b))) . z) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (((a '&' b) 'eqv' (a '&' b)) . z))) by BVFUNC_1:def 4

      .= ((( 'not' (((a '&' b) 'eqv' (a '&' b)) . z)) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (((a '&' b) 'eqv' (a '&' b)) . z))) by MARGREL1:def 19

      .= (( FALSE 'or' (a . z)) '&' ((( 'not' a) . z) 'or' TRUE )) by A2

      .= ((a . z) '&' ((( 'not' a) . z) 'or' TRUE ))

      .= ( TRUE '&' (a . z))

      .= TRUE by A1;

      hence thesis;

    end;

    begin

    reserve Y for non empty set;

    theorem :: BVFUNC_6:74

    for a,b,c,d be Function of Y, BOOLEAN holds (a 'imp' ((b '&' c) '&' d)) = (((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d))

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)) . x) = ((((a 'imp' b) '&' (a 'imp' c)) . x) '&' ((a 'imp' d) . x)) by MARGREL1:def 20

      .= ((((a 'imp' b) . x) '&' ((a 'imp' c) . x)) '&' ((a 'imp' d) . x)) by MARGREL1:def 20

      .= (((( 'not' (a . x)) 'or' (b . x)) '&' ((a 'imp' c) . x)) '&' ((a 'imp' d) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (a . x)) 'or' (c . x))) '&' ((a 'imp' d) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (b . x)) '&' (( 'not' (a . x)) 'or' (c . x))) '&' (( 'not' (a . x)) 'or' (d . x))) by BVFUNC_1:def 8

      .= ((( 'not' (a . x)) 'or' ((b . x) '&' (c . x))) '&' (( 'not' (a . x)) 'or' (d . x))) by XBOOLEAN: 9

      .= (( 'not' (a . x)) 'or' (((b . x) '&' (c . x)) '&' (d . x))) by XBOOLEAN: 9

      .= (( 'not' (a . x)) 'or' (((b '&' c) . x) '&' (d . x))) by MARGREL1:def 20

      .= (( 'not' (a . x)) 'or' (((b '&' c) '&' d) . x)) by MARGREL1:def 20

      .= ((a 'imp' ((b '&' c) '&' d)) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:75

    for a,b,c,d be Function of Y, BOOLEAN holds (a 'imp' ((b 'or' c) 'or' d)) = (((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d))

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d)) . x) = ((((a 'imp' b) 'or' (a 'imp' c)) . x) 'or' ((a 'imp' d) . x)) by BVFUNC_1:def 4

      .= ((((a 'imp' b) . x) 'or' ((a 'imp' c) . x)) 'or' ((a 'imp' d) . x)) by BVFUNC_1:def 4

      .= (((( 'not' (a . x)) 'or' (b . x)) 'or' ((a 'imp' c) . x)) 'or' ((a 'imp' d) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (b . x)) 'or' (( 'not' (a . x)) 'or' (c . x))) 'or' ((a 'imp' d) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (b . x)) 'or' (( 'not' (a . x)) 'or' (c . x))) 'or' (( 'not' (a . x)) 'or' (d . x))) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (( 'not' (a . x)) 'or' (b . x))) 'or' (c . x)) 'or' (( 'not' (a . x)) 'or' (d . x)))

      .= ((((( 'not' (a . x)) 'or' ( 'not' (a . x))) 'or' (b . x)) 'or' (c . x)) 'or' (( 'not' (a . x)) 'or' (d . x))) by BINARITH: 11

      .= ((( 'not' (a . x)) 'or' ((b . x) 'or' (c . x))) 'or' (( 'not' (a . x)) 'or' (d . x)))

      .= ((( 'not' (a . x)) 'or' ((b 'or' c) . x)) 'or' (( 'not' (a . x)) 'or' (d . x))) by BVFUNC_1:def 4

      .= ((( 'not' (a . x)) 'or' (( 'not' (a . x)) 'or' ((b 'or' c) . x))) 'or' (d . x))

      .= (((( 'not' (a . x)) 'or' ( 'not' (a . x))) 'or' ((b 'or' c) . x)) 'or' (d . x)) by BINARITH: 11

      .= (( 'not' (a . x)) 'or' (((b 'or' c) . x) 'or' (d . x)))

      .= (( 'not' (a . x)) 'or' (((b 'or' c) 'or' d) . x)) by BVFUNC_1:def 4

      .= ((a 'imp' ((b 'or' c) 'or' d)) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:76

    for a,b,c,d be Function of Y, BOOLEAN holds (((a '&' b) '&' c) 'imp' d) = (((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d))

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)) . x) = ((((a 'imp' d) 'or' (b 'imp' d)) . x) 'or' ((c 'imp' d) . x)) by BVFUNC_1:def 4

      .= ((((a 'imp' d) . x) 'or' ((b 'imp' d) . x)) 'or' ((c 'imp' d) . x)) by BVFUNC_1:def 4

      .= (((( 'not' (a . x)) 'or' (d . x)) 'or' ((b 'imp' d) . x)) 'or' ((c 'imp' d) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (d . x)) 'or' (( 'not' (b . x)) 'or' (d . x))) 'or' ((c 'imp' d) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (d . x)) 'or' (( 'not' (b . x)) 'or' (d . x))) 'or' (( 'not' (c . x)) 'or' (d . x))) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' ((d . x) 'or' ( 'not' (b . x)))) 'or' (d . x)) 'or' (( 'not' (c . x)) 'or' (d . x)))

      .= ((((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' (d . x)) 'or' (d . x)) 'or' (( 'not' (c . x)) 'or' (d . x)))

      .= (((( 'not' (a . x)) 'or' ( 'not' (b . x))) 'or' ((d . x) 'or' (d . x))) 'or' (( 'not' (c . x)) 'or' (d . x))) by BINARITH: 11

      .= ((( 'not' ((a . x) '&' (b . x))) 'or' ((d . x) 'or' ( 'not' (c . x)))) 'or' (d . x))

      .= (((( 'not' ((a . x) '&' (b . x))) 'or' ( 'not' (c . x))) 'or' (d . x)) 'or' (d . x))

      .= ((( 'not' ((a . x) '&' (b . x))) 'or' ( 'not' (c . x))) 'or' ((d . x) 'or' (d . x))) by BINARITH: 11

      .= (( 'not' (((a '&' b) . x) '&' (c . x))) 'or' (d . x)) by MARGREL1:def 20

      .= (( 'not' (((a '&' b) '&' c) . x)) 'or' (d . x)) by MARGREL1:def 20

      .= ((((a '&' b) '&' c) 'imp' d) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:77

    for a,b,c,d be Function of Y, BOOLEAN holds (((a 'or' b) 'or' c) 'imp' d) = (((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d))

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)) . x) = ((((a 'imp' d) '&' (b 'imp' d)) . x) '&' ((c 'imp' d) . x)) by MARGREL1:def 20

      .= ((((a 'imp' d) . x) '&' ((b 'imp' d) . x)) '&' ((c 'imp' d) . x)) by MARGREL1:def 20

      .= (((( 'not' (a . x)) 'or' (d . x)) '&' ((b 'imp' d) . x)) '&' ((c 'imp' d) . x)) by BVFUNC_1:def 8

      .= (((( 'not' (a . x)) 'or' (d . x)) '&' (( 'not' (b . x)) 'or' (d . x))) '&' ((c 'imp' d) . x)) by BVFUNC_1:def 8

      .= ((((d . x) 'or' ( 'not' (a . x))) '&' (( 'not' (b . x)) 'or' (d . x))) '&' (( 'not' (c . x)) 'or' (d . x))) by BVFUNC_1:def 8

      .= ((( 'not' ((a . x) 'or' (b . x))) 'or' (d . x)) '&' (( 'not' (c . x)) 'or' (d . x))) by XBOOLEAN: 9

      .= (((d . x) 'or' ( 'not' ((a 'or' b) . x))) '&' (( 'not' (c . x)) 'or' (d . x))) by BVFUNC_1:def 4

      .= (( 'not' (((a 'or' b) . x) 'or' (c . x))) 'or' (d . x)) by XBOOLEAN: 9

      .= (( 'not' (((a 'or' b) 'or' c) . x)) 'or' (d . x)) by BVFUNC_1:def 4

      .= ((((a 'or' b) 'or' c) 'imp' d) . x) by BVFUNC_1:def 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:78

    for a,b,c be Function of Y, BOOLEAN holds (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) = (((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) = (((( 'not' a) 'or' b) '&' (b 'imp' c)) '&' (c 'imp' a)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (c 'imp' a)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' a)) by BVFUNC_4: 8

      .= (((( 'not' a) '&' (( 'not' b) 'or' c)) 'or' (b '&' (( 'not' b) 'or' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 12

      .= (((( 'not' a) '&' (( 'not' b) 'or' c)) 'or' ((b '&' ( 'not' b)) 'or' (b '&' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 12

      .= (((( 'not' a) '&' (( 'not' b) 'or' c)) 'or' (( O_el Y) 'or' (b '&' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_4: 5

      .= (((( 'not' a) '&' (( 'not' b) 'or' c)) 'or' (b '&' c)) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 9

      .= (((( 'not' a) 'or' (b '&' c)) '&' ((( 'not' b) 'or' c) 'or' (b '&' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 11

      .= ((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((( 'not' b) 'or' c) 'or' (b '&' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 11

      .= ((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (((( 'not' b) 'or' c) 'or' b) '&' ((( 'not' b) 'or' c) 'or' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 11

      .= ((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((c 'or' (( 'not' b) 'or' b)) '&' ((( 'not' b) 'or' c) 'or' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 8

      .= ((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((c 'or' ( I_el Y)) '&' ((( 'not' b) 'or' c) 'or' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_4: 6

      .= ((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (( I_el Y) '&' ((( 'not' b) 'or' c) 'or' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 10

      .= ((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((( 'not' b) 'or' c) 'or' c)) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 6

      .= ((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (( 'not' b) 'or' (c 'or' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 8

      .= (((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((( 'not' b) 'or' c) '&' (( 'not' c) 'or' a))) by BVFUNC_1: 4

      .= (((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((( 'not' b) '&' (( 'not' c) 'or' a)) 'or' (c '&' (( 'not' c) 'or' a)))) by BVFUNC_1: 12

      .= (((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((( 'not' b) '&' (( 'not' c) 'or' a)) 'or' ((c '&' ( 'not' c)) 'or' (c '&' a)))) by BVFUNC_1: 12

      .= (((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((( 'not' b) '&' (( 'not' c) 'or' a)) 'or' (( O_el Y) 'or' (c '&' a)))) by BVFUNC_4: 5

      .= (((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((( 'not' b) '&' (( 'not' c) 'or' a)) 'or' (c '&' a))) by BVFUNC_1: 9

      .= (((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' ((( 'not' b) 'or' (c '&' a)) '&' ((( 'not' c) 'or' a) 'or' (c '&' a)))) by BVFUNC_1: 11

      .= (((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (((( 'not' b) 'or' c) '&' (( 'not' b) 'or' a)) '&' ((( 'not' c) 'or' a) 'or' (c '&' a)))) by BVFUNC_1: 11

      .= (((( 'not' a) 'or' c) '&' (( 'not' a) 'or' b)) '&' (((( 'not' b) 'or' a) '&' (( 'not' b) 'or' c)) '&' (((( 'not' c) 'or' a) 'or' c) '&' ((( 'not' c) 'or' a) 'or' a)))) by BVFUNC_1: 11

      .= (((( 'not' a) 'or' c) '&' (( 'not' a) 'or' b)) '&' (((( 'not' b) 'or' a) '&' (( 'not' b) 'or' c)) '&' ((a 'or' (( 'not' c) 'or' c)) '&' ((( 'not' c) 'or' a) 'or' a)))) by BVFUNC_1: 8

      .= (((( 'not' a) 'or' c) '&' (( 'not' a) 'or' b)) '&' (((( 'not' b) 'or' a) '&' (( 'not' b) 'or' c)) '&' ((a 'or' ( I_el Y)) '&' ((( 'not' c) 'or' a) 'or' a)))) by BVFUNC_4: 6

      .= (((( 'not' a) 'or' c) '&' (( 'not' a) 'or' b)) '&' (((( 'not' b) 'or' a) '&' (( 'not' b) 'or' c)) '&' (( I_el Y) '&' ((( 'not' c) 'or' a) 'or' a)))) by BVFUNC_1: 10

      .= (((( 'not' a) 'or' c) '&' (( 'not' a) 'or' b)) '&' (((( 'not' b) 'or' a) '&' (( 'not' b) 'or' c)) '&' ((( 'not' c) 'or' a) 'or' a))) by BVFUNC_1: 6

      .= (((( 'not' a) 'or' c) '&' (( 'not' a) 'or' b)) '&' (((( 'not' b) 'or' a) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' (a 'or' a)))) by BVFUNC_1: 8

      .= ((((( 'not' a) 'or' c) '&' (( 'not' a) 'or' b)) '&' ((( 'not' b) 'or' a) '&' (( 'not' b) 'or' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 4

      .= (((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (( 'not' b) 'or' a)) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 4

      .= (((((( 'not' b) 'or' a) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' b)) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 4

      .= ((((( 'not' b) 'or' a) '&' (( 'not' a) 'or' c)) '&' ((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c))) '&' (( 'not' c) 'or' a)) by BVFUNC_1: 4

      .= (((( 'not' b) 'or' a) '&' (( 'not' a) 'or' c)) '&' (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' a))) by BVFUNC_1: 4

      .= (((( 'not' b) 'or' a) '&' (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' a))) '&' (( 'not' a) 'or' c)) by BVFUNC_1: 4

      .= (((((a 'imp' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' a)) '&' (( 'not' b) 'or' a)) '&' (( 'not' a) 'or' c)) by BVFUNC_4: 8

      .= (((((a 'imp' b) '&' (b 'imp' c)) '&' (( 'not' c) 'or' a)) '&' (( 'not' b) 'or' a)) '&' (( 'not' a) 'or' c)) by BVFUNC_4: 8

      .= (((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (( 'not' b) 'or' a)) '&' (( 'not' a) 'or' c)) by BVFUNC_4: 8

      .= (((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (( 'not' a) 'or' c)) by BVFUNC_4: 8

      .= (((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c)) by BVFUNC_4: 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:79

    for a,b be Function of Y, BOOLEAN holds a = ((a '&' b) 'or' (a '&' ( 'not' b)))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a '&' b) 'or' (a '&' ( 'not' b))) . x) = ((a '&' (b 'or' ( 'not' b))) . x) by BVFUNC_1: 12

      .= ((a '&' ( I_el Y)) . x) by BVFUNC_4: 6

      .= (a . x) by BVFUNC_1: 6;

      hence thesis;

    end;

    theorem :: BVFUNC_6:80

    for a,b be Function of Y, BOOLEAN holds a = ((a 'or' b) '&' (a 'or' ( 'not' b)))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'or' b) '&' (a 'or' ( 'not' b))) . x) = ((a 'or' (b '&' ( 'not' b))) . x) by BVFUNC_1: 11

      .= ((a 'or' ( O_el Y)) . x) by BVFUNC_4: 5

      .= (a . x) by BVFUNC_1: 9;

      hence thesis;

    end;

    theorem :: BVFUNC_6:81

    for a,b,c be Function of Y, BOOLEAN holds a = (((((a '&' b) '&' c) 'or' ((a '&' b) '&' ( 'not' c))) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' ( 'not' b)) '&' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((((((a '&' b) '&' c) 'or' ((a '&' b) '&' ( 'not' c))) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' ( 'not' b)) '&' ( 'not' c))) . x) = (((((a '&' b) '&' (c 'or' ( 'not' c))) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' ( 'not' b)) '&' ( 'not' c))) . x) by BVFUNC_1: 12

      .= (((((a '&' b) '&' ( I_el Y)) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' ( 'not' b)) '&' ( 'not' c))) . x) by BVFUNC_4: 6

      .= ((((a '&' b) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' ( 'not' b)) '&' ( 'not' c))) . x) by BVFUNC_1: 6

      .= (((a '&' b) 'or' (((a '&' ( 'not' b)) '&' c) 'or' ((a '&' ( 'not' b)) '&' ( 'not' c)))) . x) by BVFUNC_1: 8

      .= (((a '&' b) 'or' ((a '&' ( 'not' b)) '&' (c 'or' ( 'not' c)))) . x) by BVFUNC_1: 12

      .= (((a '&' b) 'or' ((a '&' ( 'not' b)) '&' ( I_el Y))) . x) by BVFUNC_4: 6

      .= (((a '&' b) 'or' (a '&' ( 'not' b))) . x) by BVFUNC_1: 6

      .= ((a '&' (b 'or' ( 'not' b))) . x) by BVFUNC_1: 12

      .= ((a '&' ( I_el Y)) . x) by BVFUNC_4: 6

      .= (a . x) by BVFUNC_1: 6;

      hence thesis;

    end;

    theorem :: BVFUNC_6:82

    for a,b,c be Function of Y, BOOLEAN holds a = (((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ( 'not' c))) '&' ((a 'or' ( 'not' b)) 'or' c)) '&' ((a 'or' ( 'not' b)) 'or' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ( 'not' c))) '&' ((a 'or' ( 'not' b)) 'or' c)) '&' ((a 'or' ( 'not' b)) 'or' ( 'not' c))) . x) = (((((a 'or' b) 'or' (c '&' ( 'not' c))) '&' ((a 'or' ( 'not' b)) 'or' c)) '&' ((a 'or' ( 'not' b)) 'or' ( 'not' c))) . x) by BVFUNC_1: 11

      .= (((((a 'or' b) 'or' ( O_el Y)) '&' ((a 'or' ( 'not' b)) 'or' c)) '&' ((a 'or' ( 'not' b)) 'or' ( 'not' c))) . x) by BVFUNC_4: 5

      .= ((((a 'or' b) '&' ((a 'or' ( 'not' b)) 'or' c)) '&' ((a 'or' ( 'not' b)) 'or' ( 'not' c))) . x) by BVFUNC_1: 9

      .= (((a 'or' b) '&' (((a 'or' ( 'not' b)) 'or' c) '&' ((a 'or' ( 'not' b)) 'or' ( 'not' c)))) . x) by BVFUNC_1: 4

      .= (((a 'or' b) '&' ((a 'or' ( 'not' b)) 'or' (c '&' ( 'not' c)))) . x) by BVFUNC_1: 11

      .= (((a 'or' b) '&' ((a 'or' ( 'not' b)) 'or' ( O_el Y))) . x) by BVFUNC_4: 5

      .= (((a 'or' b) '&' (a 'or' ( 'not' b))) . x) by BVFUNC_1: 9

      .= ((a 'or' (b '&' ( 'not' b))) . x) by BVFUNC_1: 11

      .= ((a 'or' ( O_el Y)) . x) by BVFUNC_4: 5

      .= (a . x) by BVFUNC_1: 9;

      hence thesis;

    end;

    theorem :: BVFUNC_6:83

    for a,b be Function of Y, BOOLEAN holds (a '&' b) = (a '&' (( 'not' a) 'or' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a '&' (( 'not' a) 'or' b)) . x) = ((a . x) '&' ((( 'not' a) 'or' b) . x)) by MARGREL1:def 20

      .= ((a . x) '&' ((( 'not' a) . x) 'or' (b . x))) by BVFUNC_1:def 4

      .= (((a . x) '&' (( 'not' a) . x)) 'or' ((a . x) '&' (b . x))) by XBOOLEAN: 8

      .= (((a . x) '&' ( 'not' (a . x))) 'or' ((a . x) '&' (b . x))) by MARGREL1:def 19

      .= ( FALSE 'or' ((a . x) '&' (b . x))) by XBOOLEAN: 138

      .= ((a . x) '&' (b . x))

      .= ((a '&' b) . x) by MARGREL1:def 20;

      hence thesis;

    end;

    theorem :: BVFUNC_6:84

    for a,b be Function of Y, BOOLEAN holds (a 'or' b) = (a 'or' (( 'not' a) '&' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'or' (( 'not' a) '&' b)) . x) = ((a . x) 'or' ((( 'not' a) '&' b) . x)) by BVFUNC_1:def 4

      .= ((a . x) 'or' ((( 'not' a) . x) '&' (b . x))) by MARGREL1:def 20

      .= (((a . x) 'or' (( 'not' a) . x)) '&' ((a . x) 'or' (b . x))) by XBOOLEAN: 9

      .= (((a . x) 'or' ( 'not' (a . x))) '&' ((a . x) 'or' (b . x))) by MARGREL1:def 19

      .= ( TRUE '&' ((a . x) 'or' (b . x))) by XBOOLEAN: 102

      .= ((a . x) 'or' (b . x))

      .= ((a 'or' b) . x) by BVFUNC_1:def 4;

      hence thesis;

    end;

    theorem :: BVFUNC_6:85

    for a,b be Function of Y, BOOLEAN holds (a 'xor' b) = ( 'not' (a 'eqv' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'xor' b) . x) = (( 'not' ( 'not' ((( 'not' a) '&' b) 'or' (a '&' ( 'not' b))))) . x) by BVFUNC_4: 9

      .= (( 'not' (( 'not' (( 'not' a) '&' b)) '&' ( 'not' (a '&' ( 'not' b))))) . x) by BVFUNC_1: 13

      .= (( 'not' ((( 'not' ( 'not' a)) 'or' ( 'not' b)) '&' ( 'not' (a '&' ( 'not' b))))) . x) by BVFUNC_1: 14

      .= (( 'not' ((a 'or' ( 'not' b)) '&' (( 'not' a) 'or' ( 'not' ( 'not' b))))) . x) by BVFUNC_1: 14

      .= (( 'not' ((b 'imp' a) '&' (( 'not' a) 'or' b))) . x) by BVFUNC_4: 8

      .= (( 'not' ((b 'imp' a) '&' (a 'imp' b))) . x) by BVFUNC_4: 8

      .= (( 'not' (a 'eqv' b)) . x) by BVFUNC_4: 7;

      hence thesis;

    end;

    theorem :: BVFUNC_6:86

    for a,b be Function of Y, BOOLEAN holds (a 'xor' b) = ((a 'or' b) '&' (( 'not' a) 'or' ( 'not' b)))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'or' b) '&' (( 'not' a) 'or' ( 'not' b))) . x) = (((a 'or' b) . x) '&' ((( 'not' a) 'or' ( 'not' b)) . x)) by MARGREL1:def 20

      .= (((a . x) 'or' (b . x)) '&' ((( 'not' a) 'or' ( 'not' b)) . x)) by BVFUNC_1:def 4

      .= (((a . x) 'or' (b . x)) '&' ((( 'not' a) . x) 'or' (( 'not' b) . x))) by BVFUNC_1:def 4

      .= (((( 'not' a) . x) '&' ((a . x) 'or' (b . x))) 'or' (((a . x) 'or' (b . x)) '&' (( 'not' b) . x))) by XBOOLEAN: 8

      .= ((((( 'not' a) . x) '&' (a . x)) 'or' ((( 'not' a) . x) '&' (b . x))) 'or' ((( 'not' b) . x) '&' ((a . x) 'or' (b . x)))) by XBOOLEAN: 8

      .= ((((( 'not' a) . x) '&' (a . x)) 'or' ((( 'not' a) . x) '&' (b . x))) 'or' (((( 'not' b) . x) '&' (a . x)) 'or' ((( 'not' b) . x) '&' (b . x)))) by XBOOLEAN: 8

      .= (((( 'not' (a . x)) '&' (a . x)) 'or' ((( 'not' a) . x) '&' (b . x))) 'or' (((( 'not' b) . x) '&' (a . x)) 'or' ((( 'not' b) . x) '&' (b . x)))) by MARGREL1:def 19

      .= (((( 'not' (a . x)) '&' (a . x)) 'or' ((( 'not' a) . x) '&' (b . x))) 'or' (((( 'not' b) . x) '&' (a . x)) 'or' (( 'not' (b . x)) '&' (b . x)))) by MARGREL1:def 19

      .= (( FALSE 'or' ((( 'not' a) . x) '&' (b . x))) 'or' (((( 'not' b) . x) '&' (a . x)) 'or' (( 'not' (b . x)) '&' (b . x)))) by XBOOLEAN: 138

      .= (( FALSE 'or' ((( 'not' a) . x) '&' (b . x))) 'or' (((( 'not' b) . x) '&' (a . x)) 'or' FALSE )) by XBOOLEAN: 138

      .= (((( 'not' a) . x) '&' (b . x)) 'or' (((( 'not' b) . x) '&' (a . x)) 'or' FALSE ))

      .= (((( 'not' a) . x) '&' (b . x)) 'or' ((a . x) '&' (( 'not' b) . x)))

      .= (((( 'not' a) '&' b) . x) 'or' ((a . x) '&' (( 'not' b) . x))) by MARGREL1:def 20

      .= (((( 'not' a) '&' b) . x) 'or' ((a '&' ( 'not' b)) . x)) by MARGREL1:def 20

      .= (((( 'not' a) '&' b) 'or' (a '&' ( 'not' b))) . x) by BVFUNC_1:def 4

      .= ((a 'xor' b) . x) by BVFUNC_4: 9;

      hence thesis;

    end;

    theorem :: BVFUNC_6:87

    for a be Function of Y, BOOLEAN holds (a 'xor' ( I_el Y)) = ( 'not' a)

    proof

      let a be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'xor' ( I_el Y)) . x) = (((( 'not' a) '&' ( I_el Y)) 'or' (a '&' ( 'not' ( I_el Y)))) . x) by BVFUNC_4: 9

      .= (((( 'not' a) '&' ( I_el Y)) 'or' (a '&' ( O_el Y))) . x) by BVFUNC_1: 2

      .= (((( 'not' a) '&' ( I_el Y)) 'or' ( O_el Y)) . x) by BVFUNC_1: 5

      .= ((( 'not' a) '&' ( I_el Y)) . x) by BVFUNC_1: 9

      .= (( 'not' a) . x) by BVFUNC_1: 6;

      hence thesis;

    end;

    theorem :: BVFUNC_6:88

    for a be Function of Y, BOOLEAN holds (a 'xor' ( O_el Y)) = a

    proof

      let a be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'xor' ( O_el Y)) . x) = (((( 'not' a) '&' ( O_el Y)) 'or' (a '&' ( 'not' ( O_el Y)))) . x) by BVFUNC_4: 9

      .= (((( 'not' a) '&' ( O_el Y)) 'or' (a '&' ( I_el Y))) . x) by BVFUNC_1: 2

      .= (((( 'not' a) '&' ( O_el Y)) 'or' a) . x) by BVFUNC_1: 6

      .= ((( O_el Y) 'or' a) . x) by BVFUNC_1: 5

      .= (a . x) by BVFUNC_1: 9;

      hence thesis;

    end;

    theorem :: BVFUNC_6:89

    for a,b be Function of Y, BOOLEAN holds (a 'xor' b) = (( 'not' a) 'xor' ( 'not' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((( 'not' a) 'xor' ( 'not' b)) . x) = (((( 'not' ( 'not' a)) '&' ( 'not' b)) 'or' (( 'not' a) '&' ( 'not' ( 'not' b)))) . x) by BVFUNC_4: 9

      .= ((a 'xor' b) . x) by BVFUNC_4: 9;

      hence thesis;

    end;

    theorem :: BVFUNC_6:90

    for a,b be Function of Y, BOOLEAN holds ( 'not' (a 'xor' b)) = (a 'xor' ( 'not' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (( 'not' (a 'xor' b)) . x) = (( 'not' ((( 'not' a) '&' b) 'or' (a '&' ( 'not' b)))) . x) by BVFUNC_4: 9

      .= ((( 'not' (( 'not' a) '&' b)) '&' ( 'not' (a '&' ( 'not' b)))) . x) by BVFUNC_1: 13

      .= (((( 'not' ( 'not' a)) 'or' ( 'not' b)) '&' ( 'not' (a '&' ( 'not' b)))) . x) by BVFUNC_1: 14

      .= (((a 'or' ( 'not' b)) '&' (( 'not' a) 'or' ( 'not' ( 'not' b)))) . x) by BVFUNC_1: 14

      .= ((((a 'or' ( 'not' b)) '&' ( 'not' a)) 'or' ((a 'or' ( 'not' b)) '&' b)) . x) by BVFUNC_1: 12

      .= ((((a '&' ( 'not' a)) 'or' (( 'not' b) '&' ( 'not' a))) 'or' ((a 'or' ( 'not' b)) '&' b)) . x) by BVFUNC_1: 12

      .= (((( O_el Y) 'or' (( 'not' b) '&' ( 'not' a))) 'or' ((a 'or' ( 'not' b)) '&' b)) . x) by BVFUNC_4: 5

      .= (((( 'not' b) '&' ( 'not' a)) 'or' ((a 'or' ( 'not' b)) '&' b)) . x) by BVFUNC_1: 9

      .= (((( 'not' b) '&' ( 'not' a)) 'or' ((a '&' b) 'or' (( 'not' b) '&' b))) . x) by BVFUNC_1: 12

      .= (((( 'not' b) '&' ( 'not' a)) 'or' ((a '&' b) 'or' ( O_el Y))) . x) by BVFUNC_4: 5

      .= (((( 'not' a) '&' ( 'not' b)) 'or' (a '&' ( 'not' ( 'not' b)))) . x) by BVFUNC_1: 9

      .= ((a 'xor' ( 'not' b)) . x) by BVFUNC_4: 9;

      hence thesis;

    end;

    theorem :: BVFUNC_6:91



     Th18: for a,b be Function of Y, BOOLEAN holds (a 'eqv' b) = ((a 'or' ( 'not' b)) '&' (( 'not' a) 'or' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a 'or' ( 'not' b)) '&' (( 'not' a) 'or' b)) . x) = (((a 'or' ( 'not' b)) '&' (a 'imp' b)) . x) by BVFUNC_4: 8

      .= (((a 'imp' b) '&' (b 'imp' a)) . x) by BVFUNC_4: 8

      .= ((a 'eqv' b) . x) by BVFUNC_4: 7;

      hence thesis;

    end;

    theorem :: BVFUNC_6:92

    for a,b be Function of Y, BOOLEAN holds (a 'eqv' b) = ((a '&' b) 'or' (( 'not' a) '&' ( 'not' b)))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (((a '&' b) 'or' (( 'not' a) '&' ( 'not' b))) . x) = ((((a '&' b) 'or' ( 'not' a)) '&' ((a '&' b) 'or' ( 'not' b))) . x) by BVFUNC_1: 11

      .= ((((a 'or' ( 'not' a)) '&' (b 'or' ( 'not' a))) '&' ((a '&' b) 'or' ( 'not' b))) . x) by BVFUNC_1: 11

      .= ((((a 'or' ( 'not' a)) '&' (b 'or' ( 'not' a))) '&' ((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' b)))) . x) by BVFUNC_1: 11

      .= (((( I_el Y) '&' (b 'or' ( 'not' a))) '&' ((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' b)))) . x) by BVFUNC_4: 6

      .= (((( I_el Y) '&' (b 'or' ( 'not' a))) '&' ((a 'or' ( 'not' b)) '&' ( I_el Y))) . x) by BVFUNC_4: 6

      .= (((b 'or' ( 'not' a)) '&' ((a 'or' ( 'not' b)) '&' ( I_el Y))) . x) by BVFUNC_1: 6

      .= (((b 'or' ( 'not' a)) '&' (a 'or' ( 'not' b))) . x) by BVFUNC_1: 6

      .= ((a 'eqv' b) . x) by Th18;

      hence thesis;

    end;

    theorem :: BVFUNC_6:93

    for a be Function of Y, BOOLEAN holds (a 'eqv' ( I_el Y)) = a

    proof

      let a be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'eqv' ( I_el Y)) . x) = (((a 'imp' ( I_el Y)) '&' (( I_el Y) 'imp' a)) . x) by BVFUNC_4: 7

      .= (((( 'not' a) 'or' ( I_el Y)) '&' (( I_el Y) 'imp' a)) . x) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' ( I_el Y)) '&' (( 'not' ( I_el Y)) 'or' a)) . x) by BVFUNC_4: 8

      .= ((( I_el Y) '&' (( 'not' ( I_el Y)) 'or' a)) . x) by BVFUNC_1: 10

      .= ((( I_el Y) '&' (( O_el Y) 'or' a)) . x) by BVFUNC_1: 2

      .= ((( I_el Y) '&' a) . x) by BVFUNC_1: 9

      .= (a . x) by BVFUNC_1: 6;

      hence thesis;

    end;

    theorem :: BVFUNC_6:94

    for a be Function of Y, BOOLEAN holds (a 'eqv' ( O_el Y)) = ( 'not' a)

    proof

      let a be Function of Y, BOOLEAN ;

      let x be Element of Y;

      ((a 'eqv' ( O_el Y)) . x) = (((a 'imp' ( O_el Y)) '&' (( O_el Y) 'imp' a)) . x) by BVFUNC_4: 7

      .= (((( 'not' a) 'or' ( O_el Y)) '&' (( O_el Y) 'imp' a)) . x) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' ( O_el Y)) '&' (( 'not' ( O_el Y)) 'or' a)) . x) by BVFUNC_4: 8

      .= ((( 'not' a) '&' (( 'not' ( O_el Y)) 'or' a)) . x) by BVFUNC_1: 9

      .= ((( 'not' a) '&' (( I_el Y) 'or' a)) . x) by BVFUNC_1: 2

      .= ((( 'not' a) '&' ( I_el Y)) . x) by BVFUNC_1: 10

      .= (( 'not' a) . x) by BVFUNC_1: 6;

      hence thesis;

    end;

    theorem :: BVFUNC_6:95

    for a,b be Function of Y, BOOLEAN holds ( 'not' (a 'eqv' b)) = (a 'eqv' ( 'not' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let x be Element of Y;

      (( 'not' (a 'eqv' b)) . x) = (( 'not' ((a 'imp' b) '&' (b 'imp' a))) . x) by BVFUNC_4: 7

      .= (( 'not' ((( 'not' a) 'or' b) '&' (b 'imp' a))) . x) by BVFUNC_4: 8

      .= (( 'not' ((( 'not' a) 'or' b) '&' (( 'not' b) 'or' a))) . x) by BVFUNC_4: 8

      .= ((( 'not' (( 'not' a) 'or' b)) 'or' ( 'not' (( 'not' b) 'or' a))) . x) by BVFUNC_1: 14

      .= (((( 'not' ( 'not' a)) '&' ( 'not' b)) 'or' ( 'not' (( 'not' b) 'or' a))) . x) by BVFUNC_1: 13

      .= (((a '&' ( 'not' b)) 'or' (( 'not' ( 'not' b)) '&' ( 'not' a))) . x) by BVFUNC_1: 13

      .= ((((a '&' ( 'not' b)) 'or' b) '&' ((a '&' ( 'not' b)) 'or' ( 'not' a))) . x) by BVFUNC_1: 11

      .= ((((a 'or' b) '&' (( 'not' b) 'or' b)) '&' ((a '&' ( 'not' b)) 'or' ( 'not' a))) . x) by BVFUNC_1: 11

      .= ((((a 'or' b) '&' (( 'not' b) 'or' b)) '&' ((a 'or' ( 'not' a)) '&' (( 'not' b) 'or' ( 'not' a)))) . x) by BVFUNC_1: 11

      .= ((((a 'or' b) '&' ( I_el Y)) '&' ((a 'or' ( 'not' a)) '&' (( 'not' b) 'or' ( 'not' a)))) . x) by BVFUNC_4: 6

      .= ((((a 'or' b) '&' ( I_el Y)) '&' (( I_el Y) '&' (( 'not' b) 'or' ( 'not' a)))) . x) by BVFUNC_4: 6

      .= (((a 'or' b) '&' (( I_el Y) '&' (( 'not' b) 'or' ( 'not' a)))) . x) by BVFUNC_1: 6

      .= (((( 'not' a) 'or' ( 'not' b)) '&' (( 'not' ( 'not' b)) 'or' a)) . x) by BVFUNC_1: 6

      .= (((( 'not' a) 'or' ( 'not' b)) '&' (( 'not' b) 'imp' a)) . x) by BVFUNC_4: 8

      .= (((a 'imp' ( 'not' b)) '&' (( 'not' b) 'imp' a)) . x) by BVFUNC_4: 8

      .= ((a 'eqv' ( 'not' b)) . x) by BVFUNC_4: 7;

      hence thesis;

    end;

    theorem :: BVFUNC_6:96

    for a,b be Function of Y, BOOLEAN holds ( 'not' a) '<' ((a 'imp' b) 'eqv' ( 'not' a))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let z be Element of Y;

      assume

       A1: (( 'not' a) . z) = TRUE ;

      (((a 'imp' b) 'eqv' ( 'not' a)) . z) = (((( 'not' a) 'or' b) 'eqv' ( 'not' a)) . z) by BVFUNC_4: 8

      .= ((((( 'not' a) 'or' b) 'imp' ( 'not' a)) '&' (( 'not' a) 'imp' (( 'not' a) 'or' b))) . z) by BVFUNC_4: 7

      .= (((( 'not' (( 'not' a) 'or' b)) 'or' ( 'not' a)) '&' (( 'not' a) 'imp' (( 'not' a) 'or' b))) . z) by BVFUNC_4: 8

      .= (((( 'not' (( 'not' a) 'or' b)) 'or' ( 'not' a)) '&' (( 'not' ( 'not' a)) 'or' (( 'not' a) 'or' b))) . z) by BVFUNC_4: 8

      .= (((( 'not' (( 'not' a) 'or' b)) 'or' ( 'not' a)) . z) '&' ((( 'not' ( 'not' a)) 'or' (( 'not' a) 'or' b)) . z)) by MARGREL1:def 20

      .= (((( 'not' (( 'not' a) 'or' b)) . z) 'or' (( 'not' a) . z)) '&' ((( 'not' ( 'not' a)) 'or' (( 'not' a) 'or' b)) . z)) by BVFUNC_1:def 4

      .= ((( 'not' ((( 'not' a) 'or' b) . z)) 'or' (( 'not' a) . z)) '&' ((( 'not' ( 'not' a)) 'or' (( 'not' a) 'or' b)) . z)) by MARGREL1:def 19

      .= ((( 'not' ((( 'not' a) . z) 'or' (b . z))) 'or' (( 'not' a) . z)) '&' ((( 'not' ( 'not' a)) 'or' (( 'not' a) 'or' b)) . z)) by BVFUNC_1:def 4

      .= (((( 'not' ( 'not' (a . z))) '&' ( 'not' (b . z))) 'or' (( 'not' a) . z)) '&' ((( 'not' ( 'not' a)) 'or' (( 'not' a) 'or' b)) . z)) by MARGREL1:def 19

      .= ((((a . z) '&' ( 'not' (b . z))) 'or' (( 'not' a) . z)) '&' ((( 'not' ( 'not' a)) . z) 'or' ((( 'not' a) 'or' b) . z))) by BVFUNC_1:def 4

      .= ((((a . z) '&' ( 'not' (b . z))) 'or' (( 'not' a) . z)) '&' ((a . z) 'or' ((( 'not' a) . z) 'or' (b . z)))) by BVFUNC_1:def 4

      .= ( TRUE '&' ( FALSE 'or' ( TRUE 'or' (b . z)))) by A1

      .= ( FALSE 'or' ( TRUE 'or' (b . z)))

      .= ( TRUE 'or' (b . z))

      .= TRUE ;

      hence thesis;

    end;

    theorem :: BVFUNC_6:97

    for a,b be Function of Y, BOOLEAN holds ( 'not' a) '<' ((b 'imp' a) 'eqv' ( 'not' b))

    proof

      let a,b be Function of Y, BOOLEAN ;

      let z be Element of Y;

      assume (( 'not' a) . z) = TRUE ;

      then

       A1: ( 'not' (a . z)) = TRUE by MARGREL1:def 19;

      (((b 'imp' a) 'eqv' ( 'not' b)) . z) = (((( 'not' b) 'or' a) 'eqv' ( 'not' b)) . z) by BVFUNC_4: 8

      .= ((((( 'not' b) 'or' a) 'imp' ( 'not' b)) '&' (( 'not' b) 'imp' (( 'not' b) 'or' a))) . z) by BVFUNC_4: 7

      .= (((( 'not' (( 'not' b) 'or' a)) 'or' ( 'not' b)) '&' (( 'not' b) 'imp' (( 'not' b) 'or' a))) . z) by BVFUNC_4: 8

      .= (((( 'not' (( 'not' b) 'or' a)) 'or' ( 'not' b)) '&' (( 'not' ( 'not' b)) 'or' (( 'not' b) 'or' a))) . z) by BVFUNC_4: 8

      .= (((( 'not' (( 'not' b) 'or' a)) 'or' ( 'not' b)) . z) '&' ((( 'not' ( 'not' b)) 'or' (( 'not' b) 'or' a)) . z)) by MARGREL1:def 20

      .= (((( 'not' (( 'not' b) 'or' a)) . z) 'or' (( 'not' b) . z)) '&' ((( 'not' ( 'not' b)) 'or' (( 'not' b) 'or' a)) . z)) by BVFUNC_1:def 4

      .= ((( 'not' ((( 'not' b) 'or' a) . z)) 'or' (( 'not' b) . z)) '&' ((( 'not' ( 'not' b)) 'or' (( 'not' b) 'or' a)) . z)) by MARGREL1:def 19

      .= ((( 'not' ((( 'not' b) . z) 'or' (a . z))) 'or' (( 'not' b) . z)) '&' ((( 'not' ( 'not' b)) 'or' (( 'not' b) 'or' a)) . z)) by BVFUNC_1:def 4

      .= (((( 'not' ( 'not' (b . z))) '&' ( 'not' (a . z))) 'or' (( 'not' b) . z)) '&' ((( 'not' ( 'not' b)) 'or' (( 'not' b) 'or' a)) . z)) by MARGREL1:def 19

      .= ((((b . z) '&' ( 'not' (a . z))) 'or' (( 'not' b) . z)) '&' ((( 'not' ( 'not' b)) . z) 'or' ((( 'not' b) 'or' a) . z))) by BVFUNC_1:def 4

      .= ((((b . z) '&' ( 'not' (a . z))) 'or' (( 'not' b) . z)) '&' ((( 'not' ( 'not' b)) . z) 'or' ((( 'not' b) . z) 'or' (a . z)))) by BVFUNC_1:def 4

      .= ((((b . z) '&' ( 'not' (a . z))) 'or' (( 'not' b) . z)) '&' ((b . z) 'or' (( 'not' (b . z)) 'or' (a . z)))) by MARGREL1:def 19

      .= ((( TRUE '&' (b . z)) 'or' (( 'not' b) . z)) '&' ((b . z) 'or' (( 'not' (b . z)) 'or' FALSE ))) by A1

      .= (((b . z) 'or' (( 'not' b) . z)) '&' ((b . z) 'or' (( 'not' (b . z)) 'or' FALSE )))

      .= (((b . z) 'or' ( 'not' (b . z))) '&' ((b . z) 'or' (( 'not' (b . z)) 'or' FALSE ))) by MARGREL1:def 19

      .= ( TRUE '&' ((b . z) 'or' (( 'not' (b . z)) 'or' FALSE )))

      .= ((b . z) 'or' (( 'not' (b . z)) 'or' FALSE ))

      .= (((b . z) 'or' ( 'not' (b . z))) 'or' FALSE )

      .= ( TRUE 'or' FALSE ) by XBOOLEAN: 102

      .= TRUE ;

      hence thesis;

    end;

    theorem :: BVFUNC_6:98

    for a,b be Function of Y, BOOLEAN holds a '<' (((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a)

    proof

      let a,b be Function of Y, BOOLEAN ;

      let z be Element of Y;

      assume

       A1: (a . z) = TRUE ;



       A2: (((a 'or' b) 'eqv' (b 'or' a)) . z) = ((((a 'or' b) 'imp' (a 'or' b)) '&' ((a 'or' b) 'imp' (a 'or' b))) . z) by BVFUNC_4: 7

      .= ((( 'not' (a 'or' b)) 'or' (a 'or' b)) . z) by BVFUNC_4: 8

      .= (( I_el Y) . z) by BVFUNC_4: 6

      .= TRUE by BVFUNC_1:def 11;

      ((((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z) = (((((a 'or' b) 'eqv' (a 'or' b)) 'imp' a) '&' (a 'imp' ((a 'or' b) 'eqv' (a 'or' b)))) . z) by BVFUNC_4: 7

      .= (((((a 'or' b) 'eqv' (a 'or' b)) 'imp' a) . z) '&' ((a 'imp' ((a 'or' b) 'eqv' (a 'or' b))) . z)) by MARGREL1:def 20

      .= (((( 'not' ((a 'or' b) 'eqv' (a 'or' b))) 'or' a) . z) '&' ((a 'imp' ((a 'or' b) 'eqv' (a 'or' b))) . z)) by BVFUNC_4: 8

      .= (((( 'not' ((a 'or' b) 'eqv' (a 'or' b))) 'or' a) . z) '&' ((( 'not' a) 'or' ((a 'or' b) 'eqv' (a 'or' b))) . z)) by BVFUNC_4: 8

      .= (((( 'not' ((a 'or' b) 'eqv' (a 'or' b))) . z) 'or' (a . z)) '&' ((( 'not' a) 'or' ((a 'or' b) 'eqv' (a 'or' b))) . z)) by BVFUNC_1:def 4

      .= (((( 'not' ((a 'or' b) 'eqv' (a 'or' b))) . z) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (((a 'or' b) 'eqv' (a 'or' b)) . z))) by BVFUNC_1:def 4

      .= ((( 'not' (((a 'or' b) 'eqv' (a 'or' b)) . z)) 'or' (a . z)) '&' ((( 'not' a) . z) 'or' (((a 'or' b) 'eqv' (a 'or' b)) . z))) by MARGREL1:def 19

      .= (( FALSE 'or' (a . z)) '&' ((( 'not' a) . z) 'or' TRUE )) by A2

      .= ((a . z) '&' ((( 'not' a) . z) 'or' TRUE ))

      .= ((a . z) '&' TRUE )

      .= TRUE by A1;

      hence thesis;

    end;

    theorem :: BVFUNC_6:99

    for a be Function of Y, BOOLEAN holds (a 'imp' (( 'not' a) 'eqv' ( 'not' a))) = ( I_el Y)

    proof

      let a be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((a 'imp' (( 'not' a) 'eqv' ( 'not' a))) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' (( 'not' a) 'eqv' ( 'not' a))) . x) = ((( 'not' a) 'or' (( 'not' a) 'eqv' ( 'not' a))) . x) by BVFUNC_4: 8

        .= ((( 'not' a) 'or' ((( 'not' a) 'imp' ( 'not' a)) '&' (( 'not' a) 'imp' ( 'not' a)))) . x) by BVFUNC_4: 7

        .= ((( 'not' a) 'or' (( 'not' ( 'not' a)) 'or' ( 'not' a))) . x) by BVFUNC_4: 8

        .= ((( 'not' a) 'or' ( I_el Y)) . x) by BVFUNC_4: 6

        .= TRUE by BVFUNC_1: 10, BVFUNC_1:def 11;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:100

    for a,b be Function of Y, BOOLEAN holds (((a 'imp' b) 'imp' a) 'imp' a) = ( I_el Y)

    proof

      let a,b be Function of Y, BOOLEAN ;

      for x be Element of Y holds ((((a 'imp' b) 'imp' a) 'imp' a) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a 'imp' b) 'imp' a) 'imp' a) . x) = ((( 'not' ((a 'imp' b) 'imp' a)) 'or' a) . x) by BVFUNC_4: 8

        .= ((( 'not' (( 'not' (a 'imp' b)) 'or' a)) 'or' a) . x) by BVFUNC_4: 8

        .= ((( 'not' (( 'not' (( 'not' a) 'or' b)) 'or' a)) 'or' a) . x) by BVFUNC_4: 8

        .= ((( 'not' ((( 'not' ( 'not' a)) '&' ( 'not' b)) 'or' a)) 'or' a) . x) by BVFUNC_1: 13

        .= ((( 'not' ((a 'or' a) '&' (( 'not' b) 'or' a))) 'or' a) . x) by BVFUNC_1: 11

        .= (((( 'not' a) 'or' ( 'not' (( 'not' b) 'or' a))) 'or' a) . x) by BVFUNC_1: 14

        .= (((( 'not' a) 'or' (( 'not' ( 'not' b)) '&' ( 'not' a))) 'or' a) . x) by BVFUNC_1: 13

        .= ((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' ( 'not' a))) 'or' a) . x) by BVFUNC_1: 11

        .= ((((( 'not' a) 'or' b) 'or' a) '&' (( 'not' a) 'or' a)) . x) by BVFUNC_1: 11

        .= ((((( 'not' a) 'or' b) 'or' a) '&' ( I_el Y)) . x) by BVFUNC_4: 6

        .= (((( 'not' a) 'or' b) 'or' a) . x) by BVFUNC_1: 6

        .= ((b 'or' (( 'not' a) 'or' a)) . x) by BVFUNC_1: 8

        .= ((b 'or' ( I_el Y)) . x) by BVFUNC_4: 6

        .= TRUE by BVFUNC_1: 10, BVFUNC_1:def 11;

        hence thesis;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:101

    for a,b,c,d be Function of Y, BOOLEAN holds ((((a 'imp' c) '&' (b 'imp' d)) '&' (( 'not' c) 'or' ( 'not' d))) 'imp' (( 'not' a) 'or' ( 'not' b))) = ( I_el Y)

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((((a 'imp' c) '&' (b 'imp' d)) '&' (( 'not' c) 'or' ( 'not' d))) 'imp' (( 'not' a) 'or' ( 'not' b))) . x) = TRUE

      proof

        let x be Element of Y;

        ((((a 'imp' c) '&' (b 'imp' d)) '&' (( 'not' c) 'or' ( 'not' d))) 'imp' (( 'not' a) 'or' ( 'not' b))) = (( 'not' (((a 'imp' c) '&' (b 'imp' d)) '&' (( 'not' c) 'or' ( 'not' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_4: 8

        .= (( 'not' (((( 'not' a) 'or' c) '&' (b 'imp' d)) '&' (( 'not' c) 'or' ( 'not' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_4: 8

        .= (( 'not' (((( 'not' a) 'or' c) '&' (( 'not' b) 'or' d)) '&' (( 'not' c) 'or' ( 'not' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_4: 8

        .= ((( 'not' ((( 'not' a) 'or' c) '&' (( 'not' b) 'or' d))) 'or' ( 'not' (( 'not' c) 'or' ( 'not' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 14

        .= (((( 'not' (( 'not' a) 'or' c)) 'or' ( 'not' (( 'not' b) 'or' d))) 'or' ( 'not' (( 'not' c) 'or' ( 'not' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 14

        .= ((((( 'not' ( 'not' a)) '&' ( 'not' c)) 'or' ( 'not' (( 'not' b) 'or' d))) 'or' ( 'not' (( 'not' c) 'or' ( 'not' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 13

        .= ((((a '&' ( 'not' c)) 'or' (( 'not' ( 'not' b)) '&' ( 'not' d))) 'or' ( 'not' (( 'not' c) 'or' ( 'not' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 13

        .= ((((a '&' ( 'not' c)) 'or' (b '&' ( 'not' d))) 'or' (( 'not' ( 'not' c)) '&' ( 'not' ( 'not' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 13

        .= (((a '&' ( 'not' c)) 'or' ((b '&' ( 'not' d)) 'or' (c '&' d))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 8

        .= (((a '&' ( 'not' c)) 'or' ((b 'or' (c '&' d)) '&' (( 'not' d) 'or' (c '&' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 11

        .= (((a '&' ( 'not' c)) 'or' ((b 'or' (c '&' d)) '&' ((( 'not' d) 'or' c) '&' (( 'not' d) 'or' d)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 11

        .= (((a '&' ( 'not' c)) 'or' ((b 'or' (c '&' d)) '&' ((( 'not' d) 'or' c) '&' ( I_el Y)))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_4: 6

        .= (((a '&' ( 'not' c)) 'or' ((b 'or' (c '&' d)) '&' (( 'not' d) 'or' c))) 'or' (( 'not' a) 'or' ( 'not' b))) by BVFUNC_1: 6

        .= (((b 'or' (c '&' d)) '&' (( 'not' d) 'or' c)) 'or' ((a '&' ( 'not' c)) 'or' (( 'not' a) 'or' ( 'not' b)))) by BVFUNC_1: 8

        .= (((b 'or' (c '&' d)) '&' (( 'not' d) 'or' c)) 'or' ((a 'or' (( 'not' a) 'or' ( 'not' b))) '&' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b))))) by BVFUNC_1: 11

        .= (((b 'or' (c '&' d)) '&' (( 'not' d) 'or' c)) 'or' (((a 'or' ( 'not' a)) 'or' ( 'not' b)) '&' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b))))) by BVFUNC_1: 8

        .= (((b 'or' (c '&' d)) '&' (( 'not' d) 'or' c)) 'or' ((( I_el Y) 'or' ( 'not' b)) '&' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b))))) by BVFUNC_4: 6

        .= (((b 'or' (c '&' d)) '&' (( 'not' d) 'or' c)) 'or' (( I_el Y) '&' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b))))) by BVFUNC_1: 10

        .= (((b 'or' (c '&' d)) '&' (( 'not' d) 'or' c)) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b)))) by BVFUNC_1: 6

        .= (((b 'or' (c '&' d)) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b)))) '&' ((( 'not' d) 'or' c) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b))))) by BVFUNC_1: 11

        .= (((b 'or' (c '&' d)) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b)))) '&' (((( 'not' d) 'or' c) 'or' ( 'not' c)) 'or' (( 'not' a) 'or' ( 'not' b)))) by BVFUNC_1: 8

        .= (((b 'or' (c '&' d)) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b)))) '&' ((( 'not' d) 'or' (c 'or' ( 'not' c))) 'or' (( 'not' a) 'or' ( 'not' b)))) by BVFUNC_1: 8

        .= (((b 'or' (c '&' d)) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b)))) '&' ((( 'not' d) 'or' ( I_el Y)) 'or' (( 'not' a) 'or' ( 'not' b)))) by BVFUNC_4: 6

        .= (((b 'or' (c '&' d)) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b)))) '&' (( I_el Y) 'or' (( 'not' a) 'or' ( 'not' b)))) by BVFUNC_1: 10

        .= (((b 'or' (c '&' d)) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b)))) '&' ( I_el Y)) by BVFUNC_1: 10

        .= ((b 'or' (c '&' d)) 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b)))) by BVFUNC_1: 6

        .= ((c '&' d) 'or' (b 'or' (( 'not' c) 'or' (( 'not' a) 'or' ( 'not' b))))) by BVFUNC_1: 8

        .= ((c '&' d) 'or' ((b 'or' (( 'not' b) 'or' ( 'not' a))) 'or' ( 'not' c))) by BVFUNC_1: 8

        .= ((c '&' d) 'or' (((b 'or' ( 'not' b)) 'or' ( 'not' a)) 'or' ( 'not' c))) by BVFUNC_1: 8

        .= ((c '&' d) 'or' ((( I_el Y) 'or' ( 'not' a)) 'or' ( 'not' c))) by BVFUNC_4: 6

        .= ((c '&' d) 'or' (( I_el Y) 'or' ( 'not' c))) by BVFUNC_1: 10

        .= ((c '&' d) 'or' ( I_el Y)) by BVFUNC_1: 10

        .= ( I_el Y) by BVFUNC_1: 10;

        hence thesis by BVFUNC_1:def 11;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    theorem :: BVFUNC_6:102

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for x be Element of Y holds (((a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))) . x) = TRUE

      proof

        let x be Element of Y;

        ((a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))) = (( 'not' (a 'imp' b)) 'or' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))) by BVFUNC_4: 8

        .= (( 'not' (( 'not' a) 'or' b)) 'or' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))) by BVFUNC_4: 8

        .= (( 'not' (( 'not' a) 'or' b)) 'or' ((( 'not' a) 'or' (b 'imp' c)) 'imp' (a 'imp' c))) by BVFUNC_4: 8

        .= (( 'not' (( 'not' a) 'or' b)) 'or' ((( 'not' a) 'or' (( 'not' b) 'or' c)) 'imp' (a 'imp' c))) by BVFUNC_4: 8

        .= (( 'not' (( 'not' a) 'or' b)) 'or' ((( 'not' a) 'or' (( 'not' b) 'or' c)) 'imp' (( 'not' a) 'or' c))) by BVFUNC_4: 8

        .= (( 'not' (( 'not' a) 'or' b)) 'or' (( 'not' (( 'not' a) 'or' (( 'not' b) 'or' c))) 'or' (( 'not' a) 'or' c))) by BVFUNC_4: 8

        .= ((( 'not' ( 'not' a)) '&' ( 'not' b)) 'or' (( 'not' (( 'not' a) 'or' (( 'not' b) 'or' c))) 'or' (( 'not' a) 'or' c))) by BVFUNC_1: 13

        .= ((( 'not' ( 'not' a)) '&' ( 'not' b)) 'or' ((( 'not' ( 'not' a)) '&' ( 'not' (( 'not' b) 'or' c))) 'or' (( 'not' a) 'or' c))) by BVFUNC_1: 13

        .= ((a '&' ( 'not' b)) 'or' ((( 'not' ( 'not' a)) '&' (( 'not' ( 'not' b)) '&' ( 'not' c))) 'or' (( 'not' a) 'or' c))) by BVFUNC_1: 13

        .= ((a '&' ( 'not' b)) 'or' ((a 'or' (( 'not' a) 'or' c)) '&' ((b '&' ( 'not' c)) 'or' (( 'not' a) 'or' c)))) by BVFUNC_1: 11

        .= ((a '&' ( 'not' b)) 'or' (((a 'or' ( 'not' a)) 'or' c) '&' ((b '&' ( 'not' c)) 'or' (( 'not' a) 'or' c)))) by BVFUNC_1: 8

        .= ((a '&' ( 'not' b)) 'or' ((( I_el Y) 'or' c) '&' ((b '&' ( 'not' c)) 'or' (( 'not' a) 'or' c)))) by BVFUNC_4: 6

        .= ((a '&' ( 'not' b)) 'or' (( I_el Y) '&' ((b '&' ( 'not' c)) 'or' (( 'not' a) 'or' c)))) by BVFUNC_1: 10

        .= ((a '&' ( 'not' b)) 'or' ((b '&' ( 'not' c)) 'or' (( 'not' a) 'or' c))) by BVFUNC_1: 6

        .= ((a '&' ( 'not' b)) 'or' ((b 'or' (( 'not' a) 'or' c)) '&' (( 'not' c) 'or' (( 'not' a) 'or' c)))) by BVFUNC_1: 11

        .= ((a '&' ( 'not' b)) 'or' ((b 'or' (( 'not' a) 'or' c)) '&' ((( 'not' c) 'or' c) 'or' ( 'not' a)))) by BVFUNC_1: 8

        .= ((a '&' ( 'not' b)) 'or' ((b 'or' (( 'not' a) 'or' c)) '&' (( I_el Y) 'or' ( 'not' a)))) by BVFUNC_4: 6

        .= ((a '&' ( 'not' b)) 'or' ((b 'or' (( 'not' a) 'or' c)) '&' ( I_el Y))) by BVFUNC_1: 10

        .= ((a '&' ( 'not' b)) 'or' (b 'or' (( 'not' a) 'or' c))) by BVFUNC_1: 6

        .= ((a 'or' (b 'or' (( 'not' a) 'or' c))) '&' (( 'not' b) 'or' (b 'or' (( 'not' a) 'or' c)))) by BVFUNC_1: 11

        .= ((a 'or' (b 'or' (( 'not' a) 'or' c))) '&' ((( 'not' b) 'or' b) 'or' (( 'not' a) 'or' c))) by BVFUNC_1: 8

        .= ((a 'or' (b 'or' (( 'not' a) 'or' c))) '&' (( I_el Y) 'or' (( 'not' a) 'or' c))) by BVFUNC_4: 6

        .= ((a 'or' (b 'or' (( 'not' a) 'or' c))) '&' ( I_el Y)) by BVFUNC_1: 10

        .= (a 'or' (b 'or' (( 'not' a) 'or' c))) by BVFUNC_1: 6

        .= (a 'or' ((( 'not' a) 'or' b) 'or' c)) by BVFUNC_1: 8

        .= ((a 'or' (( 'not' a) 'or' b)) 'or' c) by BVFUNC_1: 8

        .= (((a 'or' ( 'not' a)) 'or' b) 'or' c) by BVFUNC_1: 8

        .= ((( I_el Y) 'or' b) 'or' c) by BVFUNC_4: 6

        .= (( I_el Y) 'or' c) by BVFUNC_1: 10

        .= ( I_el Y) by BVFUNC_1: 10;

        hence thesis by BVFUNC_1:def 11;

      end;

      hence thesis by BVFUNC_1:def 11;

    end;

    begin

    reserve Y for non empty set,

a,b,c,d,e,f,g for Function of Y, BOOLEAN ;



     Lm1: (a '&' b) '<' a

    proof

      let x be Element of Y;

      assume ((a '&' b) . x) = TRUE ;

      then ((a . x) '&' (b . x)) = TRUE by MARGREL1:def 20;

      hence thesis by MARGREL1: 12;

    end;



     Lm2: ((a '&' b) '&' c) '<' a & ((a '&' b) '&' c) '<' b

    proof

      ((a '&' b) '&' c) = ((c '&' b) '&' a) & (((c '&' b) '&' a) 'imp' a) = ( I_el Y) by BVFUNC_1: 4, Th38;

      hence ((a '&' b) '&' c) '<' a by BVFUNC_1: 16;

      ((a '&' b) '&' c) = ((a '&' c) '&' b) & (((a '&' c) '&' b) 'imp' b) = ( I_el Y) by BVFUNC_1: 4, Th38;

      hence thesis by BVFUNC_1: 16;

    end;



     Lm3: (((a '&' b) '&' c) '&' d) '<' a & (((a '&' b) '&' c) '&' d) '<' b

    proof



       A1: ((((d '&' c) '&' b) '&' a) 'imp' a) = ( I_el Y) by Th38;

      (((a '&' b) '&' c) '&' d) = ((d '&' c) '&' (b '&' a)) by BVFUNC_1: 4

      .= (((d '&' c) '&' b) '&' a) by BVFUNC_1: 4;

      hence (((a '&' b) '&' c) '&' d) '<' a by A1, BVFUNC_1: 16;



       A2: ((((a '&' c) '&' d) '&' b) 'imp' b) = ( I_el Y) by Th38;

      (((a '&' b) '&' c) '&' d) = (((a '&' c) '&' b) '&' d) by BVFUNC_1: 4

      .= (((a '&' c) '&' d) '&' b) by BVFUNC_1: 4;

      hence thesis by A2, BVFUNC_1: 16;

    end;



     Lm4: ((((a '&' b) '&' c) '&' d) '&' e) '<' a & ((((a '&' b) '&' c) '&' d) '&' e) '<' b

    proof



       A1: (((((e '&' d) '&' c) '&' b) '&' a) 'imp' a) = ( I_el Y) by Th38;

      ((((a '&' b) '&' c) '&' d) '&' e) = ((e '&' d) '&' (c '&' (b '&' a))) by BVFUNC_1: 4

      .= (((e '&' d) '&' c) '&' (b '&' a)) by BVFUNC_1: 4

      .= ((((e '&' d) '&' c) '&' b) '&' a) by BVFUNC_1: 4;

      hence ((((a '&' b) '&' c) '&' d) '&' e) '<' a by A1, BVFUNC_1: 16;



       A2: (((((a '&' c) '&' d) '&' e) '&' b) 'imp' b) = ( I_el Y) by Th38;

      ((((a '&' b) '&' c) '&' d) '&' e) = ((((a '&' c) '&' b) '&' d) '&' e) by BVFUNC_1: 4

      .= ((((a '&' c) '&' d) '&' b) '&' e) by BVFUNC_1: 4

      .= ((((a '&' c) '&' d) '&' e) '&' b) by BVFUNC_1: 4;

      hence thesis by A2, BVFUNC_1: 16;

    end;



     Lm5: (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '<' a & (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '<' b

    proof



       A1: ((((((f '&' e) '&' d) '&' c) '&' b) '&' a) 'imp' a) = ( I_el Y) by Th38;

      (((((a '&' b) '&' c) '&' d) '&' e) '&' f) = ((f '&' e) '&' (d '&' (c '&' (b '&' a)))) by BVFUNC_1: 4

      .= (((f '&' e) '&' d) '&' (c '&' (b '&' a))) by BVFUNC_1: 4

      .= ((((f '&' e) '&' d) '&' c) '&' (b '&' a)) by BVFUNC_1: 4

      .= (((((f '&' e) '&' d) '&' c) '&' b) '&' a) by BVFUNC_1: 4;

      hence (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '<' a by A1, BVFUNC_1: 16;



       A2: ((((((f '&' e) '&' d) '&' c) '&' a) '&' b) 'imp' b) = ( I_el Y) by Th38;

      (((((a '&' b) '&' c) '&' d) '&' e) '&' f) = ((f '&' e) '&' (d '&' (c '&' (b '&' a)))) by BVFUNC_1: 4

      .= (((f '&' e) '&' d) '&' (c '&' (b '&' a))) by BVFUNC_1: 4

      .= ((((f '&' e) '&' d) '&' c) '&' (b '&' a)) by BVFUNC_1: 4

      .= (((((f '&' e) '&' d) '&' c) '&' a) '&' b) by BVFUNC_1: 4;

      hence thesis by A2, BVFUNC_1: 16;

    end;



     Lm6: ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) '<' a & ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) '<' b

    proof



       A1: (((((((g '&' f) '&' e) '&' d) '&' c) '&' b) '&' a) 'imp' a) = ( I_el Y) by Th38;

      ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) = ((g '&' f) '&' (e '&' (d '&' (c '&' (b '&' a))))) by BVFUNC_1: 4

      .= (((g '&' f) '&' e) '&' (d '&' (c '&' (b '&' a)))) by BVFUNC_1: 4

      .= ((((g '&' f) '&' e) '&' d) '&' (c '&' (b '&' a))) by BVFUNC_1: 4

      .= (((((g '&' f) '&' e) '&' d) '&' c) '&' (b '&' a)) by BVFUNC_1: 4

      .= ((((((g '&' f) '&' e) '&' d) '&' c) '&' b) '&' a) by BVFUNC_1: 4;

      hence ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) '<' a by A1, BVFUNC_1: 16;



       A2: (((((((a '&' g) '&' f) '&' e) '&' d) '&' c) '&' b) 'imp' b) = ( I_el Y) by Th38;

      ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) = (((((a '&' (c '&' b)) '&' d) '&' e) '&' f) '&' g) by BVFUNC_1: 4

      .= ((((a '&' (d '&' (c '&' b))) '&' e) '&' f) '&' g) by BVFUNC_1: 4

      .= (((a '&' (e '&' (d '&' (c '&' b)))) '&' f) '&' g) by BVFUNC_1: 4

      .= ((a '&' (f '&' (e '&' (d '&' (c '&' b))))) '&' g) by BVFUNC_1: 4

      .= ((a '&' g) '&' (f '&' (e '&' (d '&' (c '&' b))))) by BVFUNC_1: 4

      .= (((a '&' g) '&' f) '&' (e '&' (d '&' (c '&' b)))) by BVFUNC_1: 4

      .= ((((a '&' g) '&' f) '&' e) '&' (d '&' (c '&' b))) by BVFUNC_1: 4

      .= (((((a '&' g) '&' f) '&' e) '&' d) '&' (c '&' b)) by BVFUNC_1: 4

      .= ((((((a '&' g) '&' f) '&' e) '&' d) '&' c) '&' b) by BVFUNC_1: 4;

      hence thesis by A2, BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:103



     Th1: ((a 'or' b) '&' (b 'imp' c)) '<' (a 'or' c)

    proof

      let z be Element of Y;



       A1: (((a 'or' b) '&' (b 'imp' c)) . z) = (((a 'or' b) . z) '&' ((b 'imp' c) . z)) by MARGREL1:def 20

      .= (((a 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_4: 8

      .= (((a 'or' b) . z) '&' ((( 'not' b) . z) 'or' (c . z))) by BVFUNC_1:def 4

      .= (((a . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'or' b) '&' (b 'imp' c)) . z) = TRUE ;

      now

        assume ((a 'or' c) . z) <> TRUE ;

        then ((a 'or' c) . z) = FALSE by XBOOLEAN:def 3;

        then

         A3: ((a . z) 'or' (c . z)) = FALSE by BVFUNC_1:def 4;

        (c . z) = TRUE or (c . z) = FALSE by XBOOLEAN:def 3;



        then (((a . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) = ((b . z) '&' ( 'not' (b . z))) by A3, MARGREL1:def 19

        .= FALSE by XBOOLEAN: 138;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:104



     Th2: (a '&' (a 'imp' b)) '<' b

    proof

      let z be Element of Y;



       A1: ((a '&' (a 'imp' b)) . z) = ((a . z) '&' ((a 'imp' b) . z)) by MARGREL1:def 20

      .= ((a . z) '&' ((( 'not' a) 'or' b) . z)) by BVFUNC_4: 8

      .= ((a . z) '&' ((( 'not' a) . z) 'or' (b . z))) by BVFUNC_1:def 4

      .= (((a . z) '&' (( 'not' a) . z)) 'or' ((a . z) '&' (b . z))) by XBOOLEAN: 8

      .= (((a . z) '&' ( 'not' (a . z))) 'or' ((a . z) '&' (b . z))) by MARGREL1:def 19

      .= ( FALSE 'or' ((a . z) '&' (b . z))) by XBOOLEAN: 138

      .= ((a . z) '&' (b . z));

      assume

       A2: ((a '&' (a 'imp' b)) . z) = TRUE ;

      now

        assume (b . z) <> TRUE ;



        then ((a . z) '&' (b . z)) = ( FALSE '&' (a . z)) by XBOOLEAN:def 3

        .= FALSE ;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:105

    ((a 'imp' b) '&' ( 'not' b)) '<' ( 'not' a)

    proof

      let z be Element of Y;

      reconsider bz = (b . z) as boolean object;



       A1: (((a 'imp' b) '&' ( 'not' b)) . z) = (((a 'imp' b) . z) '&' (( 'not' b) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' (( 'not' b) . z)) by BVFUNC_4: 8

      .= ((( 'not' b) . z) '&' ((( 'not' a) . z) 'or' (b . z))) by BVFUNC_1:def 4

      .= (((( 'not' b) . z) '&' (( 'not' a) . z)) 'or' ((( 'not' b) . z) '&' (b . z))) by XBOOLEAN: 8

      .= (((( 'not' b) . z) '&' (( 'not' a) . z)) 'or' (( 'not' bz) '&' bz)) by MARGREL1:def 19

      .= (((( 'not' b) . z) '&' (( 'not' a) . z)) 'or' FALSE ) by XBOOLEAN: 138

      .= ((( 'not' b) . z) '&' (( 'not' a) . z));

      assume

       A2: (((a 'imp' b) '&' ( 'not' b)) . z) = TRUE ;

      now

        assume (( 'not' a) . z) <> TRUE ;



        then ((( 'not' b) . z) '&' (( 'not' a) . z)) = ( FALSE '&' (( 'not' b) . z)) by XBOOLEAN:def 3

        .= FALSE ;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:106

    ((a 'or' b) '&' ( 'not' a)) '<' b

    proof

      let z be Element of Y;

      reconsider az = (a . z) as boolean object;



       A1: (((a 'or' b) '&' ( 'not' a)) . z) = (((a 'or' b) . z) '&' (( 'not' a) . z)) by MARGREL1:def 20

      .= ((( 'not' a) . z) '&' ((a . z) 'or' (b . z))) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) '&' (a . z)) 'or' ((( 'not' a) . z) '&' (b . z))) by XBOOLEAN: 8

      .= ((( 'not' az) '&' az) 'or' ((( 'not' a) . z) '&' (b . z))) by MARGREL1:def 19

      .= ( FALSE 'or' ((( 'not' a) . z) '&' (b . z))) by XBOOLEAN: 138

      .= ((( 'not' a) . z) '&' (b . z));

      assume

       A2: (((a 'or' b) '&' ( 'not' a)) . z) = TRUE ;

      now

        assume (b . z) <> TRUE ;



        then ((( 'not' a) . z) '&' (b . z)) = ( FALSE '&' (( 'not' a) . z)) by XBOOLEAN:def 3

        .= FALSE ;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:107

    ((a 'imp' b) '&' (( 'not' a) 'imp' b)) '<' b

    proof

      let z be Element of Y;

      reconsider az = (a . z) as boolean object;



       A1: (((a 'imp' b) '&' (( 'not' a) 'imp' b)) . z) = (((a 'imp' b) . z) '&' ((( 'not' a) 'imp' b) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' a) 'imp' b) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' ( 'not' a)) 'or' b) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((a 'or' b) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((a . z) 'or' (b . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (( 'not' a) 'imp' b)) . z) = TRUE ;

      now

        assume (b . z) <> TRUE ;

        then (b . z) = FALSE by XBOOLEAN:def 3;



        then (((( 'not' a) . z) 'or' (b . z)) '&' ((a . z) 'or' (b . z))) = (( 'not' az) '&' az) by MARGREL1:def 19

        .= FALSE by XBOOLEAN: 138;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:108

    ((a 'imp' b) '&' (a 'imp' ( 'not' b))) '<' ( 'not' a)

    proof

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (a 'imp' ( 'not' b))) . z) = (((a 'imp' b) . z) '&' ((a 'imp' ( 'not' b)) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((a 'imp' ( 'not' b)) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' a) 'or' ( 'not' b)) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' a) 'or' ( 'not' b)) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' a) . z) 'or' (( 'not' b) . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (a 'imp' ( 'not' b))) . z) = TRUE ;

      now

        assume (( 'not' a) . z) <> TRUE ;

        then (( 'not' a) . z) = FALSE by XBOOLEAN:def 3;



        then (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' a) . z) 'or' (( 'not' b) . z))) = ((b . z) '&' ( 'not' (b . z))) by MARGREL1:def 19

        .= FALSE by XBOOLEAN: 138;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:109

    (a 'imp' (b '&' c)) '<' (a 'imp' b)

    proof

      let z be Element of Y;



       A1: ((a 'imp' (b '&' c)) . z) = ((( 'not' a) 'or' (b '&' c)) . z) by BVFUNC_4: 8

      .= ((( 'not' a) . z) 'or' ((b '&' c) . z)) by BVFUNC_1:def 4

      .= ((( 'not' a) . z) 'or' ((b . z) '&' (c . z))) by MARGREL1:def 20;

      assume

       A2: ((a 'imp' (b '&' c)) . z) = TRUE ;

      now

        assume ((a 'imp' b) . z) <> TRUE ;

        then ((a 'imp' b) . z) = FALSE by XBOOLEAN:def 3;

        then ((( 'not' a) 'or' b) . z) = FALSE by BVFUNC_4: 8;

        then

         A3: ((( 'not' a) . z) 'or' (b . z)) = FALSE by BVFUNC_1:def 4;

        ((( 'not' a) . z) 'or' ((b . z) '&' (c . z))) = (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' a) . z) 'or' (c . z))) by XBOOLEAN: 9

        .= FALSE by A3;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:110

    ((a 'or' b) 'imp' c) '<' (a 'imp' c)

    proof

      let z be Element of Y;



       A1: (((a 'or' b) 'imp' c) . z) = ((( 'not' (a 'or' b)) 'or' c) . z) by BVFUNC_4: 8

      .= (((( 'not' a) '&' ( 'not' b)) 'or' c) . z) by BVFUNC_1: 13

      .= (((( 'not' a) '&' ( 'not' b)) . z) 'or' (c . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) '&' (( 'not' b) . z)) 'or' (c . z)) by MARGREL1:def 20;

      assume

       A2: (((a 'or' b) 'imp' c) . z) = TRUE ;

      now

        assume ((a 'imp' c) . z) <> TRUE ;

        then ((a 'imp' c) . z) = FALSE by XBOOLEAN:def 3;

        then ((( 'not' a) 'or' c) . z) = FALSE by BVFUNC_4: 8;

        then

         A3: ((( 'not' a) . z) 'or' (c . z)) = FALSE by BVFUNC_1:def 4;

        (((( 'not' a) . z) '&' (( 'not' b) . z)) 'or' (c . z)) = (((c . z) 'or' (( 'not' a) . z)) '&' ((c . z) 'or' (( 'not' b) . z))) by XBOOLEAN: 9

        .= FALSE by A3;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:111

    (a 'imp' b) '<' ((a '&' c) 'imp' b)

    proof

      let z be Element of Y;



       A1: ((a 'imp' b) . z) = ((( 'not' a) 'or' b) . z) by BVFUNC_4: 8

      .= ((( 'not' a) . z) 'or' (b . z)) by BVFUNC_1:def 4;

      assume

       A2: ((a 'imp' b) . z) = TRUE ;

      now

        assume (((a '&' c) 'imp' b) . z) <> TRUE ;

        then (((a '&' c) 'imp' b) . z) = FALSE by XBOOLEAN:def 3;

        then ((( 'not' (a '&' c)) 'or' b) . z) = FALSE by BVFUNC_4: 8;

        then ((( 'not' (a '&' c)) . z) 'or' (b . z)) = FALSE by BVFUNC_1:def 4;

        then (((( 'not' a) 'or' ( 'not' c)) . z) 'or' (b . z)) = FALSE by BVFUNC_1: 14;

        then (((( 'not' c) . z) 'or' (( 'not' a) . z)) 'or' (b . z)) = FALSE by BVFUNC_1:def 4;

        then ((( 'not' c) . z) 'or' ((( 'not' a) . z) 'or' (b . z))) = FALSE ;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:112

    (a 'imp' b) '<' ((a '&' c) 'imp' (b '&' c))

    proof

      let z be Element of Y;



       A1: ((a 'imp' b) . z) = ((( 'not' a) 'or' b) . z) by BVFUNC_4: 8

      .= ((( 'not' a) . z) 'or' (b . z)) by BVFUNC_1:def 4;

      assume

       A2: ((a 'imp' b) . z) = TRUE ;

      now

        assume (((a '&' c) 'imp' (b '&' c)) . z) <> TRUE ;

        then (((a '&' c) 'imp' (b '&' c)) . z) = FALSE by XBOOLEAN:def 3;

        then ((( 'not' (a '&' c)) 'or' (b '&' c)) . z) = FALSE by BVFUNC_4: 8;

        then ((( 'not' (a '&' c)) . z) 'or' ((b '&' c) . z)) = FALSE by BVFUNC_1:def 4;

        then (((( 'not' a) 'or' ( 'not' c)) . z) 'or' ((b '&' c) . z)) = FALSE by BVFUNC_1: 14;

        then (((( 'not' c) . z) 'or' (( 'not' a) . z)) 'or' ((b '&' c) . z)) = FALSE by BVFUNC_1:def 4;

        then ((( 'not' c) . z) 'or' ((( 'not' a) . z) 'or' ((b '&' c) . z))) = FALSE ;

        then

         A3: ((( 'not' c) . z) 'or' ((( 'not' a) . z) 'or' ((b . z) '&' (c . z)))) = FALSE by MARGREL1:def 20;

        ((( 'not' c) . z) 'or' (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' a) . z) 'or' (c . z)))) = (((( 'not' c) . z) 'or' (c . z)) 'or' (( 'not' a) . z)) by A2, A1

        .= ((( 'not' (c . z)) 'or' (c . z)) 'or' (( 'not' a) . z)) by MARGREL1:def 19

        .= ( TRUE 'or' (( 'not' a) . z)) by XBOOLEAN: 102

        .= TRUE ;

        hence contradiction by A3, XBOOLEAN: 9;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:113

    (a 'imp' b) '<' (a 'imp' (b 'or' c))

    proof

      let z be Element of Y;



       A1: ((a 'imp' b) . z) = ((( 'not' a) 'or' b) . z) by BVFUNC_4: 8

      .= ((( 'not' a) . z) 'or' (b . z)) by BVFUNC_1:def 4;

      assume

       A2: ((a 'imp' b) . z) = TRUE ;

      now

        assume

         A3: ((a 'imp' (b 'or' c)) . z) <> TRUE ;

        ((a 'imp' (b 'or' c)) . z) = ((( 'not' a) 'or' (b 'or' c)) . z) by BVFUNC_4: 8

        .= ((( 'not' a) . z) 'or' ((b 'or' c) . z)) by BVFUNC_1:def 4

        .= ((( 'not' a) . z) 'or' ((b . z) 'or' (c . z))) by BVFUNC_1:def 4

        .= (((( 'not' a) . z) 'or' (b . z)) 'or' (c . z))

        .= TRUE by A2, A1;

        hence contradiction by A3;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:114

    (a 'imp' b) '<' ((a 'or' c) 'imp' (b 'or' c))

    proof

      let z be Element of Y;



       A1: ((a 'imp' b) . z) = ((( 'not' a) 'or' b) . z) by BVFUNC_4: 8

      .= ((( 'not' a) . z) 'or' (b . z)) by BVFUNC_1:def 4;

      assume

       A2: ((a 'imp' b) . z) = TRUE ;

      now

        assume

         A3: (((a 'or' c) 'imp' (b 'or' c)) . z) <> TRUE ;

        (((a 'or' c) 'imp' (b 'or' c)) . z) = ((( 'not' (a 'or' c)) 'or' (b 'or' c)) . z) by BVFUNC_4: 8

        .= ((( 'not' (a 'or' c)) . z) 'or' ((b 'or' c) . z)) by BVFUNC_1:def 4

        .= ((( 'not' (a 'or' c)) . z) 'or' ((b . z) 'or' (c . z))) by BVFUNC_1:def 4

        .= (((( 'not' (a 'or' c)) . z) 'or' (b . z)) 'or' (c . z))

        .= ((((( 'not' a) '&' ( 'not' c)) . z) 'or' (b . z)) 'or' (c . z)) by BVFUNC_1: 13

        .= (((b . z) 'or' ((( 'not' a) . z) '&' (( 'not' c) . z))) 'or' (c . z)) by MARGREL1:def 20

        .= ((((( 'not' a) . z) 'or' (b . z)) '&' ((b . z) 'or' (( 'not' c) . z))) 'or' (c . z)) by XBOOLEAN: 9

        .= ((b . z) 'or' ((( 'not' c) . z) 'or' (c . z))) by A2, A1

        .= ((b . z) 'or' (( 'not' (c . z)) 'or' (c . z))) by MARGREL1:def 19

        .= ((b . z) 'or' TRUE ) by XBOOLEAN: 102

        .= TRUE ;

        hence contradiction by A3;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:115

    ((a '&' b) 'or' c) '<' (a 'or' c)

    proof

      let z be Element of Y;



       A1: (((a '&' b) 'or' c) . z) = (((a '&' b) . z) 'or' (c . z)) by BVFUNC_1:def 4

      .= (((a . z) '&' (b . z)) 'or' (c . z)) by MARGREL1:def 20;

      assume

       A2: (((a '&' b) 'or' c) . z) = TRUE ;

      now

        assume ((a 'or' c) . z) <> TRUE ;

        then ((a 'or' c) . z) = FALSE by XBOOLEAN:def 3;

        then

         A3: ((a . z) 'or' (c . z)) = FALSE by BVFUNC_1:def 4;

        (((a . z) '&' (b . z)) 'or' (c . z)) = (((c . z) 'or' (a . z)) '&' ((c . z) 'or' (b . z))) by XBOOLEAN: 9

        .= FALSE by A3;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:116

    ((a '&' b) 'or' (c '&' d)) '<' (a 'or' c)

    proof

      let z be Element of Y;



       A1: (((a '&' b) 'or' (c '&' d)) . z) = (((a '&' b) . z) 'or' ((c '&' d) . z)) by BVFUNC_1:def 4

      .= (((a . z) '&' (b . z)) 'or' ((c '&' d) . z)) by MARGREL1:def 20

      .= (((a . z) '&' (b . z)) 'or' ((c . z) '&' (d . z))) by MARGREL1:def 20;

      assume

       A2: (((a '&' b) 'or' (c '&' d)) . z) = TRUE ;

      now

        assume ((a 'or' c) . z) <> TRUE ;

        then ((a 'or' c) . z) = FALSE by XBOOLEAN:def 3;

        then

         A3: ((a . z) 'or' (c . z)) = FALSE by BVFUNC_1:def 4;

        (((a . z) '&' (b . z)) 'or' ((c . z) '&' (d . z))) = (((c . z) 'or' ((a . z) '&' (b . z))) '&' (((a . z) '&' (b . z)) 'or' (d . z))) by XBOOLEAN: 9

        .= ((((a . z) 'or' (c . z)) '&' ((c . z) 'or' (b . z))) '&' (((a . z) '&' (b . z)) 'or' (d . z))) by XBOOLEAN: 9

        .= FALSE by A3;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:117

    ((a 'imp' b) '&' (( 'not' a) 'imp' c)) '<' (b 'or' c)

    proof

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (( 'not' a) 'imp' c)) . z) = (((( 'not' a) 'or' b) '&' (( 'not' a) 'imp' c)) . z) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) '&' (( 'not' ( 'not' a)) 'or' c)) . z) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((a 'or' c) . z)) by MARGREL1:def 20

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((a 'or' c) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((a . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (( 'not' a) 'imp' c)) . z) = TRUE ;

      now

        reconsider az = (a . z) as boolean object;

        assume ((b 'or' c) . z) <> TRUE ;

        then ((b 'or' c) . z) = FALSE by XBOOLEAN:def 3;

        then

         A3: ((b . z) 'or' (c . z)) = FALSE by BVFUNC_1:def 4;

        (c . z) = TRUE or (c . z) = FALSE by XBOOLEAN:def 3;



        then (((( 'not' a) . z) 'or' (b . z)) '&' ((a . z) 'or' (c . z))) = (( 'not' az) '&' az) by A3, MARGREL1:def 19

        .= FALSE by XBOOLEAN: 138;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:118

    ((a 'imp' c) '&' (b 'imp' ( 'not' c))) '<' (( 'not' a) 'or' ( 'not' b))

    proof

      let z be Element of Y;



       A1: (((a 'imp' c) '&' (b 'imp' ( 'not' c))) . z) = (((a 'imp' c) . z) '&' ((b 'imp' ( 'not' c)) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' c) . z) '&' ((b 'imp' ( 'not' c)) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' c) . z) '&' ((( 'not' b) 'or' ( 'not' c)) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' b) 'or' ( 'not' c)) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' b) . z) 'or' (( 'not' c) . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' c) '&' (b 'imp' ( 'not' c))) . z) = TRUE ;

      now

        assume ((( 'not' a) 'or' ( 'not' b)) . z) <> TRUE ;

        then ((( 'not' a) 'or' ( 'not' b)) . z) = FALSE by XBOOLEAN:def 3;

        then

         A3: ((( 'not' a) . z) 'or' (( 'not' b) . z)) = FALSE by BVFUNC_1:def 4;

        (( 'not' b) . z) = TRUE or (( 'not' b) . z) = FALSE by XBOOLEAN:def 3;



        then (((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' b) . z) 'or' (( 'not' c) . z))) = ((c . z) '&' ( 'not' (c . z))) by A3, MARGREL1:def 19

        .= FALSE by XBOOLEAN: 138;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:119

    ((a 'or' b) '&' (( 'not' a) 'or' c)) '<' (b 'or' c)

    proof

      let z be Element of Y;



       A1: (((a 'or' b) '&' (( 'not' a) 'or' c)) . z) = (((a 'or' b) . z) '&' ((( 'not' a) 'or' c) . z)) by MARGREL1:def 20

      .= (((a . z) 'or' (b . z)) '&' ((( 'not' a) 'or' c) . z)) by BVFUNC_1:def 4

      .= (((a . z) 'or' (b . z)) '&' ((( 'not' a) . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'or' b) '&' (( 'not' a) 'or' c)) . z) = TRUE ;

      now

        assume ((b 'or' c) . z) <> TRUE ;

        then ((b 'or' c) . z) = FALSE by XBOOLEAN:def 3;

        then

         A3: ((b . z) 'or' (c . z)) = FALSE by BVFUNC_1:def 4;

        (c . z) = TRUE or (c . z) = FALSE by XBOOLEAN:def 3;



        then (((a . z) 'or' (b . z)) '&' ((( 'not' a) . z) 'or' (c . z))) = ((a . z) '&' ( 'not' (a . z))) by A3, MARGREL1:def 19

        .= FALSE by XBOOLEAN: 138;

        hence contradiction by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:120



     Th918: ((a 'imp' b) '&' (c 'imp' d)) '<' ((a '&' c) 'imp' (b '&' d))

    proof

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (c 'imp' d)) . z) = (((a 'imp' b) . z) '&' ((c 'imp' d) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((c 'imp' d) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' c) 'or' d) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' c) 'or' d) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' c) . z) 'or' (d . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (c 'imp' d)) . z) = TRUE ;

      now



         A3: (( 'not' c) . z) = TRUE or (( 'not' c) . z) = FALSE by XBOOLEAN:def 3;



         A4: ((( 'not' a) . z) 'or' (( 'not' c) . z)) = TRUE or ((( 'not' a) . z) 'or' (( 'not' c) . z)) = FALSE by XBOOLEAN:def 3;



         A5: ((b . z) '&' (d . z)) = TRUE or ((b . z) '&' (d . z)) = FALSE by XBOOLEAN:def 3;



         A6: (((a '&' c) 'imp' (b '&' d)) . z) = ((( 'not' (a '&' c)) 'or' (b '&' d)) . z) by BVFUNC_4: 8

        .= ((( 'not' (a '&' c)) . z) 'or' ((b '&' d) . z)) by BVFUNC_1:def 4

        .= (((( 'not' a) 'or' ( 'not' c)) . z) 'or' ((b '&' d) . z)) by BVFUNC_1: 14

        .= (((( 'not' a) . z) 'or' (( 'not' c) . z)) 'or' ((b '&' d) . z)) by BVFUNC_1:def 4

        .= (((( 'not' a) . z) 'or' (( 'not' c) . z)) 'or' ((b . z) '&' (d . z))) by MARGREL1:def 20;

        assume

         A7: (((a '&' c) 'imp' (b '&' d)) . z) <> TRUE ;

        now

          per cases by A7, A6, A5, MARGREL1: 12;

            case (b . z) = FALSE ;

            thus thesis by A2, A1, A6, A4, A3;

          end;

            case (d . z) = FALSE ;

            thus thesis by A2, A1, A6, A4, A3;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:121

    ((a 'imp' b) '&' (a 'imp' c)) '<' (a 'imp' (b '&' c))

    proof

      ((a 'imp' b) '&' (a 'imp' c)) '<' ((a '&' a) 'imp' (b '&' c)) by Th918;

      hence thesis;

    end;

    theorem :: BVFUNC_6:122



     Th20: ((a 'imp' c) '&' (b 'imp' c)) '<' ((a 'or' b) 'imp' c)

    proof

      (((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)) = ( I_el Y) by Th9;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:123



     Th21: ((a 'imp' b) '&' (c 'imp' d)) '<' ((a 'or' c) 'imp' (b 'or' d))

    proof

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (c 'imp' d)) . z) = (((( 'not' a) 'or' b) '&' (c 'imp' d)) . z) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) '&' (( 'not' c) 'or' d)) . z) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' c) 'or' d) . z)) by MARGREL1:def 20

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' c) 'or' d) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' c) . z) 'or' (d . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (c 'imp' d)) . z) = TRUE ;

      now



         A3: (d . z) = TRUE or (d . z) = FALSE by XBOOLEAN:def 3;



         A4: ((b . z) 'or' (d . z)) = TRUE or ((b . z) 'or' (d . z)) = FALSE by XBOOLEAN:def 3;



         A5: ((( 'not' a) . z) '&' (( 'not' c) . z)) = TRUE or ((( 'not' a) . z) '&' (( 'not' c) . z)) = FALSE by XBOOLEAN:def 3;



         A6: (((a 'or' c) 'imp' (b 'or' d)) . z) = ((( 'not' (a 'or' c)) 'or' (b 'or' d)) . z) by BVFUNC_4: 8

        .= (((( 'not' a) '&' ( 'not' c)) 'or' (b 'or' d)) . z) by BVFUNC_1: 13

        .= (((( 'not' a) '&' ( 'not' c)) . z) 'or' ((b 'or' d) . z)) by BVFUNC_1:def 4

        .= (((( 'not' a) . z) '&' (( 'not' c) . z)) 'or' ((b 'or' d) . z)) by MARGREL1:def 20

        .= (((( 'not' a) . z) '&' (( 'not' c) . z)) 'or' ((b . z) 'or' (d . z))) by BVFUNC_1:def 4;

        assume

         A7: (((a 'or' c) 'imp' (b 'or' d)) . z) <> TRUE ;

        now

          per cases by A7, A6, A5, MARGREL1: 12;

            case (( 'not' a) . z) = FALSE ;

            thus thesis by A2, A1, A6, A4, A3;

          end;

            case (( 'not' c) . z) = FALSE ;

            thus thesis by A2, A1, A6, A4, A3;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:124

    ((a 'imp' b) '&' (a 'imp' c)) '<' (a 'imp' (b 'or' c))

    proof

      ((a 'imp' b) '&' (a 'imp' c)) '<' ((a 'or' a) 'imp' (b 'or' c)) by Th21;

      hence thesis;

    end;

    theorem :: BVFUNC_6:125



     Th23: for a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN holds (((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '<' (a2 'imp' a1)

    proof

      let a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN ;



       A1: (((((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ((b2 'or' c2) 'imp' ( 'not' a2))) '&' ((b1 'or' c1) 'imp' ( 'not' a2))) 'imp' ((b1 'or' c1) 'imp' ( 'not' a2))) = ( I_el Y) by Th38;



       A2: ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' (c1 'imp' c2)) = ( I_el Y) by Lm4, BVFUNC_1: 16;



       A3: (((b1 'imp' b2) '&' (c1 'imp' c2)) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2))) = ( I_el Y) by Th21, BVFUNC_1: 16;

      ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' (b1 'imp' b2)) = ( I_el Y) by Lm4, BVFUNC_1: 16;

      then ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' ((b1 'imp' b2) '&' (c1 'imp' c2))) = ( I_el Y) by A2, th18;

      then

       A4: ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2))) = ( I_el Y) by A3, BVFUNC_5: 9;



       A5: ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' ( 'not' (a2 '&' c2))) = ( I_el Y) by Lm1, BVFUNC_1: 16;

      ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' ( 'not' (a2 '&' b2))) = ( I_el Y) by Lm2, BVFUNC_1: 16;

      then ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' (( 'not' (a2 '&' b2)) '&' ( 'not' (a2 '&' c2)))) = ( I_el Y) by A5, th18;

      then

       A6: ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' (((b1 'or' c1) 'imp' (b2 'or' c2)) '&' (( 'not' (a2 '&' b2)) '&' ( 'not' (a2 '&' c2))))) = ( I_el Y) by A4, th18;

      (( 'not' (a2 '&' b2)) '&' ( 'not' (a2 '&' c2))) = ((( 'not' a2) 'or' ( 'not' b2)) '&' ( 'not' (a2 '&' c2))) by BVFUNC_1: 14

      .= ((( 'not' b2) 'or' ( 'not' a2)) '&' (( 'not' c2) 'or' ( 'not' a2))) by BVFUNC_1: 14

      .= ((b2 'imp' ( 'not' a2)) '&' (( 'not' c2) 'or' ( 'not' a2))) by BVFUNC_4: 8

      .= ((b2 'imp' ( 'not' a2)) '&' (c2 'imp' ( 'not' a2))) by BVFUNC_4: 8

      .= ((b2 'or' c2) 'imp' ( 'not' a2)) by Th75;

      then ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' ((((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ((b2 'or' c2) 'imp' ( 'not' a2))) '&' ((b1 'or' c1) 'imp' ( 'not' a2)))) = ( I_el Y) by A6, Th12;

      then

       A7: ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' ((b1 'or' c1) 'imp' ( 'not' a2))) = ( I_el Y) by A1, BVFUNC_5: 9;

      (((a1 'or' b1) 'or' c1) '&' ((b1 'or' c1) 'imp' ( 'not' a2))) = ((a1 'or' (b1 'or' c1)) '&' ((b1 'or' c1) 'imp' ( 'not' a2))) & ((a1 'or' (b1 'or' c1)) '&' ((b1 'or' c1) 'imp' ( 'not' a2))) '<' (a1 'or' ( 'not' a2)) by Th1, BVFUNC_1: 8;

      then

       A8: ((((a1 'or' b1) 'or' c1) '&' ((b1 'or' c1) 'imp' ( 'not' a2))) 'imp' (a1 'or' ( 'not' a2))) = ( I_el Y) by BVFUNC_1: 16;

      ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' ((a1 'or' b1) 'or' c1)) = ( I_el Y) by Lm3, BVFUNC_1: 16;

      then ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' (((a1 'or' b1) 'or' c1) '&' ((b1 'or' c1) 'imp' ( 'not' a2)))) = ( I_el Y) by A7, th18;

      then ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' (a1 'or' ( 'not' a2))) = ( I_el Y) by A8, BVFUNC_5: 9;

      then ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' (a2 'imp' a1)) = ( I_el Y) by BVFUNC_4: 8;

      hence thesis by BVFUNC_1: 16;

    end;



     Lm7: for a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN holds ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2)))) = ( I_el Y)

    proof

      let a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN ;



       A1: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ( 'not' (a2 '&' c2))) = ( I_el Y) by Lm2, BVFUNC_1: 16;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (a1 'imp' a2)) = ( I_el Y) & ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (b1 'imp' b2)) = ( I_el Y) by Lm6, BVFUNC_1: 16;

      then

       A2: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2))) = ( I_el Y) by th18;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((a1 'or' b1) 'or' c1)) = ( I_el Y) by Lm4, BVFUNC_1: 16;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1))) = ( I_el Y) by A2, th18;

      then

       A3: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' c2)))) = ( I_el Y) by A1, th18;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ( 'not' (b2 '&' c2))) = ( I_el Y) by Lm1, BVFUNC_1: 16;

      hence thesis by A3, th18;

    end;



     Lm8: for a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN holds ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (b2 '&' a2))) '&' ( 'not' (b2 '&' c2)))) = ( I_el Y)

    proof

      let a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN ;



       A1: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ( 'not' (a2 '&' b2))) = ( I_el Y) by Lm3, BVFUNC_1: 16;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (a1 'imp' a2)) = ( I_el Y) & ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (c1 'imp' c2)) = ( I_el Y) by Lm5, Lm6, BVFUNC_1: 16;

      then

       A2: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((a1 'imp' a2) '&' (c1 'imp' c2))) = ( I_el Y) by th18;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((a1 'or' b1) 'or' c1)) = ( I_el Y) by Lm4, BVFUNC_1: 16;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1))) = ( I_el Y) by A2, th18;

      then

       A3: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2)))) = ( I_el Y) by A1, th18;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ( 'not' (b2 '&' c2))) = ( I_el Y) by Lm1, BVFUNC_1: 16;

      hence thesis by A3, th18;

    end;



     Lm9: for a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN holds ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2)))) = ( I_el Y)

    proof

      let a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN ;



       A1: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ( 'not' (a2 '&' b2))) = ( I_el Y) by Lm3, BVFUNC_1: 16;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (b1 'imp' b2)) = ( I_el Y) & ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (c1 'imp' c2)) = ( I_el Y) by Lm5, Lm6, BVFUNC_1: 16;

      then

       A2: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((b1 'imp' b2) '&' (c1 'imp' c2))) = ( I_el Y) by th18;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((a1 'or' b1) 'or' c1)) = ( I_el Y) by Lm4, BVFUNC_1: 16;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1))) = ( I_el Y) by A2, th18;

      then

       A3: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2)))) = ( I_el Y) by A1, th18;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ( 'not' (a2 '&' c2))) = ( I_el Y) by Lm2, BVFUNC_1: 16;

      hence thesis by A3, th18;

    end;

    theorem :: BVFUNC_6:126

    for a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN holds (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) '<' (((a2 'imp' a1) '&' (b2 'imp' b1)) '&' (c2 'imp' c1))

    proof

      let a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN ;



       A1: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2)))) = ( I_el Y) by Lm9;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2)))) = ( I_el Y) & ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (b2 '&' a2))) '&' ( 'not' (b2 '&' c2)))) = ( I_el Y) by Lm7, Lm8;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2))) '&' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (b2 '&' a2))) '&' ( 'not' (b2 '&' c2))))) = ( I_el Y) by th18;

      then

       A2: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2))) '&' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (b2 '&' c2)))) '&' (((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))))) = ( I_el Y) by A1, th18;



       A3: ((((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (b2 '&' c2))) 'imp' (b2 'imp' b1)) = ( I_el Y) by Th23, BVFUNC_1: 16;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2))) '&' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (b2 '&' c2)))) '&' (((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2)))) 'imp' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (b2 '&' c2)))) = ( I_el Y) by Lm2, BVFUNC_1: 16;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (b2 '&' c2)))) = ( I_el Y) by A2, BVFUNC_5: 9;

      then

       A4: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (b2 'imp' b1)) = ( I_el Y) by A3, BVFUNC_5: 9;

      ((((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) 'imp' (a2 'imp' a1)) = ( I_el Y) by Th23, BVFUNC_1: 16;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (a2 'imp' a1)) = ( I_el Y) by A1, BVFUNC_5: 9;

      then

       A5: ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' ((a2 'imp' a1) '&' (b2 'imp' b1))) = ( I_el Y) by A4, th18;

      (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' c1) 'or' b1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2))) '<' (c2 'imp' c1) by Th23;

      then (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2))) '<' (c2 'imp' c1) by BVFUNC_1: 8;

      then

       A6: ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2))) 'imp' (c2 'imp' c1)) = ( I_el Y) by BVFUNC_1: 16;

      ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2))) '&' (((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (b2 '&' c2)))) '&' (((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2)))) 'imp' (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2)))) = ( I_el Y) by Lm2, BVFUNC_1: 16;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (c2 '&' a2))) '&' ( 'not' (c2 '&' b2)))) = ( I_el Y) by A2, BVFUNC_5: 9;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (c2 'imp' c1)) = ( I_el Y) by A6, BVFUNC_5: 9;

      then ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) 'imp' (((a2 'imp' a1) '&' (b2 'imp' b1)) '&' (c2 'imp' c1))) = ( I_el Y) by A5, th18;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:127



     Th25: for a1,b1,a2,b2 be Function of Y, BOOLEAN holds ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) 'imp' ( 'not' (a1 '&' b1))) = ( I_el Y)

    proof

      let a1,b1,a2,b2 be Function of Y, BOOLEAN ;

      for z be Element of Y st ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) . z) = TRUE holds (( 'not' (a1 '&' b1)) . z) = TRUE

      proof

        let z be Element of Y;



         A1: ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) . z) = ((((a1 'imp' a2) '&' (b1 'imp' b2)) . z) '&' (( 'not' (a2 '&' b2)) . z)) by MARGREL1:def 20

        .= ((((a1 'imp' a2) . z) '&' ((b1 'imp' b2) . z)) '&' (( 'not' (a2 '&' b2)) . z)) by MARGREL1:def 20

        .= ((((( 'not' a1) 'or' a2) . z) '&' ((b1 'imp' b2) . z)) '&' (( 'not' (a2 '&' b2)) . z)) by BVFUNC_4: 8

        .= ((((( 'not' a1) 'or' a2) . z) '&' ((( 'not' b1) 'or' b2) . z)) '&' (( 'not' (a2 '&' b2)) . z)) by BVFUNC_4: 8

        .= ((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) 'or' b2) . z)) '&' (( 'not' (a2 '&' b2)) . z)) by BVFUNC_1:def 4

        .= ((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) . z) 'or' (b2 . z))) '&' (( 'not' (a2 '&' b2)) . z)) by BVFUNC_1:def 4

        .= ((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) . z) 'or' (b2 . z))) '&' ((( 'not' a2) 'or' ( 'not' b2)) . z)) by BVFUNC_1: 14

        .= ((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) . z) 'or' (b2 . z))) '&' ((( 'not' a2) . z) 'or' (( 'not' b2) . z))) by BVFUNC_1:def 4;

        assume

         A2: ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) . z) = TRUE ;

        now



           A3: (( 'not' b1) . z) = TRUE or (( 'not' b1) . z) = FALSE by XBOOLEAN:def 3;



           A4: (( 'not' a1) . z) = TRUE or (( 'not' a1) . z) = FALSE by XBOOLEAN:def 3;



           A5: (( 'not' (a1 '&' b1)) . z) = ((( 'not' a1) 'or' ( 'not' b1)) . z) by BVFUNC_1: 14

          .= ((( 'not' a1) . z) 'or' (( 'not' b1) . z)) by BVFUNC_1:def 4;

          assume (( 'not' (a1 '&' b1)) . z) <> TRUE ;



          then ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) . z) = ((((b2 . z) '&' (a2 . z)) '&' (( 'not' a2) . z)) 'or' (((a2 . z) '&' (b2 . z)) '&' (( 'not' b2) . z))) by A1, A5, A4, A3, XBOOLEAN: 8

          .= (((b2 . z) '&' ((a2 . z) '&' (( 'not' a2) . z))) 'or' ((a2 . z) '&' ((b2 . z) '&' (( 'not' b2) . z))))

          .= (((b2 . z) '&' ((a2 . z) '&' ( 'not' (a2 . z)))) 'or' ((a2 . z) '&' ((b2 . z) '&' (( 'not' b2) . z)))) by MARGREL1:def 19

          .= (((b2 . z) '&' ((a2 . z) '&' ( 'not' (a2 . z)))) 'or' ((a2 . z) '&' ((b2 . z) '&' ( 'not' (b2 . z))))) by MARGREL1:def 19

          .= (((b2 . z) '&' FALSE ) 'or' ((a2 . z) '&' ((b2 . z) '&' ( 'not' (b2 . z))))) by XBOOLEAN: 138

          .= ( FALSE 'or' ( FALSE '&' (a2 . z))) by XBOOLEAN: 138

          .= FALSE ;

          hence contradiction by A2;

        end;

        hence thesis;

      end;

      hence thesis by BVFUNC_1: 16, BVFUNC_1:def 12;

    end;

    theorem :: BVFUNC_6:128

    for a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN holds ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) '<' ((( 'not' (a1 '&' b1)) '&' ( 'not' (a1 '&' c1))) '&' ( 'not' (b1 '&' c1)))

    proof

      let a1,b1,c1,a2,b2,c2 be Function of Y, BOOLEAN ;



       A1: ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2))))) = ( I_el Y) by Th38;



       A2: ((((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2))) 'imp' ( 'not' (a1 '&' c1))) = ( I_el Y) by Th25;

      (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) 'imp' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) = ( I_el Y) by Th38;

      then ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) = ( I_el Y) by A1, BVFUNC_5: 9;

      then

       A3: ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' ( 'not' (a1 '&' c1))) = ( I_el Y) by A2, BVFUNC_5: 9;



       A4: ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) 'imp' ( 'not' (a1 '&' b1))) = ( I_el Y) by Th25;

      ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) = ( I_el Y) & ((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2))) 'imp' ( 'not' (b1 '&' c1))) = ( I_el Y) by Th25, Th38;

      then

       A5: ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' ( 'not' (b1 '&' c1))) = ( I_el Y) by BVFUNC_5: 9;

      (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) 'imp' (((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2)))) = ( I_el Y) by Th38;

      then ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' (((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2)))) = ( I_el Y) by A1, BVFUNC_5: 9;

      then ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' ( 'not' (a1 '&' b1))) = ( I_el Y) by A4, BVFUNC_5: 9;

      then ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' (( 'not' (a1 '&' b1)) '&' ( 'not' (a1 '&' c1)))) = ( I_el Y) by A3, th18;

      then

       A6: ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) 'imp' ((( 'not' (a1 '&' b1)) '&' ( 'not' (a1 '&' c1))) '&' ( 'not' (b1 '&' c1)))) = ( I_el Y) by A5, th18;

      ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) = (((((((a1 'imp' a2) '&' (a1 'imp' a2)) '&' ((b1 'imp' b2) '&' (b1 'imp' b2))) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2)))

      .= ((((((((a1 'imp' a2) '&' (a1 'imp' a2)) '&' (b1 'imp' b2)) '&' (b1 'imp' b2)) '&' ((c1 'imp' c2) '&' (c1 'imp' c2))) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2)) '&' (b1 'imp' b2)) '&' ((c1 'imp' c2) '&' (c1 'imp' c2))) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= (((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2)) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2))) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= ((((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2))) '&' (a1 'imp' a2)) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2))) '&' ((a1 'imp' a2) '&' (c1 'imp' c2))) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2))) '&' ((a1 'imp' a2) '&' (c1 'imp' c2))) '&' ( 'not' (a2 '&' c2))) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' ( 'not' (a2 '&' b2))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2))) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' ((b1 'imp' b2) '&' (c1 'imp' c2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' ( 'not' (b2 '&' c2))) by BVFUNC_1: 4

      .= ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' ((b1 'imp' b2) '&' (c1 'imp' c2))) '&' ( 'not' (b2 '&' c2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) by BVFUNC_1: 4

      .= (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) by BVFUNC_1: 4

      .= (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ( 'not' (a2 '&' b2))) '&' (((a1 'imp' a2) '&' (c1 'imp' c2)) '&' ( 'not' (a2 '&' c2)))) '&' (((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ( 'not' (b2 '&' c2)))) by BVFUNC_1: 4;

      hence thesis by A6, BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:129

    (a '&' b) '<' a by Lm1;

    theorem :: BVFUNC_6:130

    ((a '&' b) '&' c) '<' a & ((a '&' b) '&' c) '<' b by Lm2;

    theorem :: BVFUNC_6:131

    (((a '&' b) '&' c) '&' d) '<' a & (((a '&' b) '&' c) '&' d) '<' b by Lm3;

    theorem :: BVFUNC_6:132

    ((((a '&' b) '&' c) '&' d) '&' e) '<' a & ((((a '&' b) '&' c) '&' d) '&' e) '<' b by Lm4;

    theorem :: BVFUNC_6:133

    (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '<' a & (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '<' b by Lm5;

    theorem :: BVFUNC_6:134

    ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) '<' a & ((((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g) '<' b by Lm6;

    theorem :: BVFUNC_6:135



     Th33: a '<' b & c '<' d implies (a '&' c) '<' (b '&' d)

    proof

      assume a '<' b & c '<' d;

      then (a 'imp' b) = ( I_el Y) & (c 'imp' d) = ( I_el Y) by BVFUNC_1: 16;

      then ((a '&' c) 'imp' (b '&' d)) = ( I_el Y) by tt;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:136

    (a '&' b) '<' c implies (a '&' ( 'not' c)) '<' ( 'not' b)

    proof

      assume (a '&' b) '<' c;



      then ( I_el Y) = ((a '&' b) 'imp' c) by BVFUNC_1: 16

      .= (( 'not' (a '&' b)) 'or' c) by BVFUNC_4: 8

      .= ((( 'not' a) 'or' ( 'not' b)) 'or' c) by BVFUNC_1: 14

      .= ((( 'not' a) 'or' ( 'not' ( 'not' c))) 'or' ( 'not' b)) by BVFUNC_1: 8

      .= (( 'not' (a '&' ( 'not' c))) 'or' ( 'not' b)) by BVFUNC_1: 14

      .= ((a '&' ( 'not' c)) 'imp' ( 'not' b)) by BVFUNC_4: 8;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:137

    (((a 'imp' c) '&' (b 'imp' c)) '&' (a 'or' b)) '<' c

    proof

      set K1 = ((a 'imp' c) '&' (b 'imp' c));

      K1 '<' ((a 'or' b) 'imp' c) by Th20;

      then

       A1: (K1 '&' (a 'or' b)) '<' (((a 'or' b) 'imp' c) '&' (a 'or' b)) by Th33;

      (((a 'or' b) 'imp' c) '&' (a 'or' b)) '<' c by Th2;

      hence thesis by A1, BVFUNC_1: 15;

    end;

    theorem :: BVFUNC_6:138

    (((a 'imp' c) 'or' (b 'imp' c)) '&' (a '&' b)) '<' c

    proof

      ((a 'imp' c) 'or' (b 'imp' c)) = ((a '&' b) 'imp' c) by Th76;

      hence thesis by Th2;

    end;

    theorem :: BVFUNC_6:139

    a '<' b & c '<' d implies (a 'or' c) '<' (b 'or' d)

    proof

      assume a '<' b & c '<' d;

      then (a 'imp' b) = ( I_el Y) & (c 'imp' d) = ( I_el Y) by BVFUNC_1: 16;

      then ((a 'or' c) 'imp' (b 'or' d)) = ( I_el Y) by Th22;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:140



     Th38a: a '<' (a 'or' b)

    proof

      (a 'imp' (a 'or' b)) = ( I_el Y) by Th26;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:141

    (a '&' b) '<' (a 'or' b)

    proof

      (a '&' b) '<' a & a '<' (a 'or' b) by Lm1, Th38a;

      hence thesis by BVFUNC_1: 15;

    end;

    begin

    reserve Y for non empty set;

    theorem :: BVFUNC_6:142

    for a,b,c be Function of Y, BOOLEAN holds (((a '&' b) 'or' (b '&' c)) 'or' (c '&' a)) = (((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for z be Element of Y st ((((a '&' b) 'or' (b '&' c)) 'or' (c '&' a)) . z) = TRUE holds ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) . z) = TRUE

      proof

        let z be Element of Y;



         A1: ((((a '&' b) 'or' (b '&' c)) 'or' (c '&' a)) . z) = ((((a '&' b) 'or' (b '&' c)) . z) 'or' ((c '&' a) . z)) by BVFUNC_1:def 4

        .= ((((a '&' b) . z) 'or' ((b '&' c) . z)) 'or' ((c '&' a) . z)) by BVFUNC_1:def 4

        .= ((((a . z) '&' (b . z)) 'or' ((b '&' c) . z)) 'or' ((c '&' a) . z)) by MARGREL1:def 20

        .= ((((a . z) '&' (b . z)) 'or' ((b . z) '&' (c . z))) 'or' ((c '&' a) . z)) by MARGREL1:def 20

        .= ((((a . z) '&' (b . z)) 'or' ((b . z) '&' (c . z))) 'or' ((c . z) '&' (a . z))) by MARGREL1:def 20;

        assume

         A2: ((((a '&' b) 'or' (b '&' c)) 'or' (c '&' a)) . z) = TRUE ;

        now



           A3: ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) . z) = ((((a 'or' b) '&' (b 'or' c)) . z) '&' ((c 'or' a) . z)) by MARGREL1:def 20

          .= ((((a 'or' b) . z) '&' ((b 'or' c) . z)) '&' ((c 'or' a) . z)) by MARGREL1:def 20

          .= ((((a . z) 'or' (b . z)) '&' ((b 'or' c) . z)) '&' ((c 'or' a) . z)) by BVFUNC_1:def 4

          .= ((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c 'or' a) . z)) by BVFUNC_1:def 4

          .= ((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c . z) 'or' (a . z))) by BVFUNC_1:def 4;

          assume

           A4: ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) . z) <> TRUE ;

          now

            per cases by A4, A3, MARGREL1: 12, XBOOLEAN:def 3;

              case

               A5: (((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) = FALSE ;

              now

                per cases by A5, MARGREL1: 12;

                  case

                   A6: ((a . z) 'or' (b . z)) = FALSE ;

                  (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;

                  hence thesis by A2, A1, A6;

                end;

                  case

                   A7: ((b . z) 'or' (c . z)) = FALSE ;

                  (c . z) = TRUE or (c . z) = FALSE by XBOOLEAN:def 3;

                  hence thesis by A2, A1, A7;

                end;

              end;

              hence thesis;

            end;

              case

               A8: ((c . z) 'or' (a . z)) = FALSE ;

              (a . z) = TRUE or (a . z) = FALSE by XBOOLEAN:def 3;

              hence thesis by A2, A1, A8;

            end;

          end;

          hence thesis;

        end;

        hence thesis;

      end;

      then

       A9: (((a '&' b) 'or' (b '&' c)) 'or' (c '&' a)) '<' (((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a));

      for z be Element of Y st ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) . z) = TRUE holds ((((a '&' b) 'or' (b '&' c)) 'or' (c '&' a)) . z) = TRUE

      proof

        let z be Element of Y;



         A10: ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) . z) = ((((a 'or' b) '&' (b 'or' c)) . z) '&' ((c 'or' a) . z)) by MARGREL1:def 20

        .= ((((a 'or' b) . z) '&' ((b 'or' c) . z)) '&' ((c 'or' a) . z)) by MARGREL1:def 20

        .= ((((a . z) 'or' (b . z)) '&' ((b 'or' c) . z)) '&' ((c 'or' a) . z)) by BVFUNC_1:def 4

        .= ((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c 'or' a) . z)) by BVFUNC_1:def 4

        .= ((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c . z) 'or' (a . z))) by BVFUNC_1:def 4;

        assume

         A11: ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) . z) = TRUE ;

        now



           A12: ((b . z) '&' (c . z)) = TRUE or ((b . z) '&' (c . z)) = FALSE by XBOOLEAN:def 3;



           A13: ((c . z) '&' (a . z)) = TRUE or ((c . z) '&' (a . z)) = FALSE by XBOOLEAN:def 3;



           A14: ((((a '&' b) 'or' (b '&' c)) 'or' (c '&' a)) . z) = ((((a '&' b) 'or' (b '&' c)) . z) 'or' ((c '&' a) . z)) by BVFUNC_1:def 4

          .= ((((a '&' b) . z) 'or' ((b '&' c) . z)) 'or' ((c '&' a) . z)) by BVFUNC_1:def 4

          .= ((((a . z) '&' (b . z)) 'or' ((b '&' c) . z)) 'or' ((c '&' a) . z)) by MARGREL1:def 20

          .= ((((a . z) '&' (b . z)) 'or' ((b . z) '&' (c . z))) 'or' ((c '&' a) . z)) by MARGREL1:def 20

          .= ((((a . z) '&' (b . z)) 'or' ((b . z) '&' (c . z))) 'or' ((c . z) '&' (a . z))) by MARGREL1:def 20;

          assume

           A15: ((((a '&' b) 'or' (b '&' c)) 'or' (c '&' a)) . z) <> TRUE ;

          now

            per cases by A15, A14, A13, A12, MARGREL1: 12, XBOOLEAN:def 3;

              case (a . z) = FALSE & (b . z) = FALSE ;

              hence thesis by A11, A10;

            end;

              case (b . z) = FALSE & (c . z) = FALSE ;

              hence thesis by A11, A10;

            end;

              case (c . z) = FALSE & (a . z) = FALSE ;

              hence thesis by A11, A10;

            end;

          end;

          hence thesis;

        end;

        hence thesis;

      end;

      then (((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '<' (((a '&' b) 'or' (b '&' c)) 'or' (c '&' a));

      hence thesis by A9, BVFUNC_1: 15;

    end;



     Lm1: for a,b,c be Function of Y, BOOLEAN holds (((a '&' ( 'not' b)) 'or' (b '&' ( 'not' c))) 'or' (c '&' ( 'not' a))) '<' (((b '&' ( 'not' a)) 'or' (c '&' ( 'not' b))) 'or' (a '&' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: ((((a '&' ( 'not' b)) 'or' (b '&' ( 'not' c))) 'or' (c '&' ( 'not' a))) . z) = ((((a '&' ( 'not' b)) 'or' (b '&' ( 'not' c))) . z) 'or' ((c '&' ( 'not' a)) . z)) by BVFUNC_1:def 4

      .= ((((a '&' ( 'not' b)) . z) 'or' ((b '&' ( 'not' c)) . z)) 'or' ((c '&' ( 'not' a)) . z)) by BVFUNC_1:def 4

      .= ((((a . z) '&' (( 'not' b) . z)) 'or' ((b '&' ( 'not' c)) . z)) 'or' ((c '&' ( 'not' a)) . z)) by MARGREL1:def 20

      .= ((((a . z) '&' (( 'not' b) . z)) 'or' ((b . z) '&' (( 'not' c) . z))) 'or' ((c '&' ( 'not' a)) . z)) by MARGREL1:def 20

      .= ((((a . z) '&' (( 'not' b) . z)) 'or' ((b . z) '&' (( 'not' c) . z))) 'or' ((c . z) '&' (( 'not' a) . z))) by MARGREL1:def 20

      .= ((((a . z) '&' ( 'not' (b . z))) 'or' ((b . z) '&' (( 'not' c) . z))) 'or' ((c . z) '&' (( 'not' a) . z))) by MARGREL1:def 19

      .= ((((a . z) '&' ( 'not' (b . z))) 'or' ((b . z) '&' ( 'not' (c . z)))) 'or' ((c . z) '&' (( 'not' a) . z))) by MARGREL1:def 19

      .= ((((a . z) '&' ( 'not' (b . z))) 'or' ((b . z) '&' ( 'not' (c . z)))) 'or' ((c . z) '&' ( 'not' (a . z)))) by MARGREL1:def 19;

      assume

       A2: ((((a '&' ( 'not' b)) 'or' (b '&' ( 'not' c))) 'or' (c '&' ( 'not' a))) . z) = TRUE ;

      now



         A3: ((a . z) '&' (( 'not' c) . z)) = TRUE or ((a . z) '&' (( 'not' c) . z)) = FALSE by XBOOLEAN:def 3;

        assume

         A4: ((((b '&' ( 'not' a)) 'or' (c '&' ( 'not' b))) 'or' (a '&' ( 'not' c))) . z) <> TRUE ;



         A5: ((c . z) '&' (( 'not' b) . z)) = TRUE or ((c . z) '&' (( 'not' b) . z)) = FALSE by XBOOLEAN:def 3;



         A6: ((((b '&' ( 'not' a)) 'or' (c '&' ( 'not' b))) 'or' (a '&' ( 'not' c))) . z) = ((((b '&' ( 'not' a)) 'or' (c '&' ( 'not' b))) . z) 'or' ((a '&' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= ((((b '&' ( 'not' a)) . z) 'or' ((c '&' ( 'not' b)) . z)) 'or' ((a '&' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= ((((b . z) '&' (( 'not' a) . z)) 'or' ((c '&' ( 'not' b)) . z)) 'or' ((a '&' ( 'not' c)) . z)) by MARGREL1:def 20

        .= ((((b . z) '&' (( 'not' a) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) 'or' ((a '&' ( 'not' c)) . z)) by MARGREL1:def 20

        .= ((((b . z) '&' (( 'not' a) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) 'or' ((a . z) '&' (( 'not' c) . z))) by MARGREL1:def 20;

        (((b . z) '&' (( 'not' a) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) = TRUE or (((b . z) '&' (( 'not' a) . z)) 'or' ((c . z) '&' (( 'not' b) . z))) = FALSE by XBOOLEAN:def 3;

        then

         A7: (b . z) = FALSE or (( 'not' a) . z) = FALSE by A4, A6, A5, MARGREL1: 12;

        now

          per cases by A4, A6, A3, MARGREL1: 12;

            case (a . z) = FALSE ;

            hence thesis by A2, A1, A6, A7, MARGREL1:def 19;

          end;

            case (( 'not' c) . z) = FALSE ;

            then

             A8: ( 'not' (c . z)) = FALSE by MARGREL1:def 19;

            then ( 'not' (b . z)) = FALSE by A4, A6, A5, MARGREL1:def 19;

            hence thesis by A2, A1, A6, A5, A8, MARGREL1:def 19;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:143

    for a,b,c be Function of Y, BOOLEAN holds (((a '&' ( 'not' b)) 'or' (b '&' ( 'not' c))) 'or' (c '&' ( 'not' a))) = (((b '&' ( 'not' a)) 'or' (c '&' ( 'not' b))) 'or' (a '&' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      (((a '&' ( 'not' b)) 'or' (b '&' ( 'not' c))) 'or' (c '&' ( 'not' a))) '<' (((b '&' ( 'not' a)) 'or' (c '&' ( 'not' b))) 'or' (a '&' ( 'not' c))) & (((b '&' ( 'not' a)) 'or' (c '&' ( 'not' b))) 'or' (a '&' ( 'not' c))) '<' (((a '&' ( 'not' b)) 'or' (b '&' ( 'not' c))) 'or' (c '&' ( 'not' a))) by Lm1;

      hence thesis by BVFUNC_1: 15;

    end;



     Lm2: for a,b,c be Function of Y, BOOLEAN holds (((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' c))) '&' (c 'or' ( 'not' a))) '<' (((b 'or' ( 'not' a)) '&' (c 'or' ( 'not' b))) '&' (a 'or' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: ((((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' c))) '&' (c 'or' ( 'not' a))) . z) = ((((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' c))) . z) '&' ((c 'or' ( 'not' a)) . z)) by MARGREL1:def 20

      .= ((((a 'or' ( 'not' b)) . z) '&' ((b 'or' ( 'not' c)) . z)) '&' ((c 'or' ( 'not' a)) . z)) by MARGREL1:def 20

      .= ((((a . z) 'or' (( 'not' b) . z)) '&' ((b 'or' ( 'not' c)) . z)) '&' ((c 'or' ( 'not' a)) . z)) by BVFUNC_1:def 4

      .= ((((a . z) 'or' (( 'not' b) . z)) '&' ((b . z) 'or' (( 'not' c) . z))) '&' ((c 'or' ( 'not' a)) . z)) by BVFUNC_1:def 4

      .= ((((a . z) 'or' (( 'not' b) . z)) '&' ((b . z) 'or' (( 'not' c) . z))) '&' ((c . z) 'or' (( 'not' a) . z))) by BVFUNC_1:def 4

      .= ((((a . z) 'or' ( 'not' (b . z))) '&' ((b . z) 'or' (( 'not' c) . z))) '&' ((c . z) 'or' (( 'not' a) . z))) by MARGREL1:def 19

      .= ((((a . z) 'or' ( 'not' (b . z))) '&' ((b . z) 'or' ( 'not' (c . z)))) '&' ((c . z) 'or' (( 'not' a) . z))) by MARGREL1:def 19

      .= ((((a . z) 'or' ( 'not' (b . z))) '&' ((b . z) 'or' ( 'not' (c . z)))) '&' ((c . z) 'or' ( 'not' (a . z)))) by MARGREL1:def 19;

      assume

       A2: ((((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' c))) '&' (c 'or' ( 'not' a))) . z) = TRUE ;

      now



         A3: ((((b 'or' ( 'not' a)) '&' (c 'or' ( 'not' b))) '&' (a 'or' ( 'not' c))) . z) = ((((b 'or' ( 'not' a)) '&' (c 'or' ( 'not' b))) . z) '&' ((a 'or' ( 'not' c)) . z)) by MARGREL1:def 20

        .= ((((b 'or' ( 'not' a)) . z) '&' ((c 'or' ( 'not' b)) . z)) '&' ((a 'or' ( 'not' c)) . z)) by MARGREL1:def 20

        .= ((((b . z) 'or' (( 'not' a) . z)) '&' ((c 'or' ( 'not' b)) . z)) '&' ((a 'or' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= ((((b . z) 'or' (( 'not' a) . z)) '&' ((c . z) 'or' (( 'not' b) . z))) '&' ((a 'or' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= ((((b . z) 'or' (( 'not' a) . z)) '&' ((c . z) 'or' (( 'not' b) . z))) '&' ((a . z) 'or' (( 'not' c) . z))) by BVFUNC_1:def 4;

        assume

         A4: ((((b 'or' ( 'not' a)) '&' (c 'or' ( 'not' b))) '&' (a 'or' ( 'not' c))) . z) <> TRUE ;

        now

          per cases by A4, A3, MARGREL1: 12, XBOOLEAN:def 3;

            case

             A5: (((b . z) 'or' (( 'not' a) . z)) '&' ((c . z) 'or' (( 'not' b) . z))) = FALSE ;

            now

              per cases by A5, MARGREL1: 12;

                case

                 A6: ((b . z) 'or' (( 'not' a) . z)) = FALSE ;



                 A7: (( 'not' a) . z) = TRUE or (( 'not' a) . z) = FALSE by XBOOLEAN:def 3;

                then ( 'not' (a . z)) = FALSE by A6, MARGREL1:def 19;

                hence thesis by A2, A1, A6, A7, XBOOLEAN: 138;

              end;

                case

                 A8: ((c . z) 'or' (( 'not' b) . z)) = FALSE ;



                 A9: (( 'not' b) . z) = TRUE or (( 'not' b) . z) = FALSE by XBOOLEAN:def 3;

                then ( 'not' (b . z)) = FALSE by A8, MARGREL1:def 19;

                hence thesis by A2, A1, A8, A9, XBOOLEAN: 138;

              end;

            end;

            hence thesis;

          end;

            case

             A10: ((a . z) 'or' (( 'not' c) . z)) = FALSE ;



             A11: (( 'not' c) . z) = TRUE or (( 'not' c) . z) = FALSE by XBOOLEAN:def 3;

            then ( 'not' (c . z)) = FALSE by A10, MARGREL1:def 19;

            hence thesis by A2, A1, A10, A11, XBOOLEAN: 138;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:144

    for a,b,c be Function of Y, BOOLEAN holds (((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' c))) '&' (c 'or' ( 'not' a))) = (((b 'or' ( 'not' a)) '&' (c 'or' ( 'not' b))) '&' (a 'or' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      (((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' c))) '&' (c 'or' ( 'not' a))) '<' (((b 'or' ( 'not' a)) '&' (c 'or' ( 'not' b))) '&' (a 'or' ( 'not' c))) & (((b 'or' ( 'not' a)) '&' (c 'or' ( 'not' b))) '&' (a 'or' ( 'not' c))) '<' (((a 'or' ( 'not' b)) '&' (b 'or' ( 'not' c))) '&' (c 'or' ( 'not' a))) by Lm2;

      hence thesis by BVFUNC_1: 15;

    end;

    theorem :: BVFUNC_6:145

    for a,b,c be Function of Y, BOOLEAN holds (c 'imp' a) = ( I_el Y) & (c 'imp' b) = ( I_el Y) implies (c 'imp' (a 'or' b)) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      assume

       A1: (c 'imp' a) = ( I_el Y) & (c 'imp' b) = ( I_el Y);

      (c 'imp' (a 'or' b)) = ((c 'imp' a) 'or' (c 'imp' b)) by Th73

      .= ( I_el Y) by A1;

      hence thesis;

    end;

    theorem :: BVFUNC_6:146

    for a,b,c be Function of Y, BOOLEAN holds (a 'imp' c) = ( I_el Y) & (b 'imp' c) = ( I_el Y) implies ((a '&' b) 'imp' c) = ( I_el Y)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      (a 'imp' c) = ( I_el Y) & (b 'imp' c) = ( I_el Y) implies ((a '&' b) 'imp' (c '&' c)) = ( I_el Y) by tt;

      hence thesis;

    end;

    theorem :: BVFUNC_6:147

    for a1,a2,b1,b2,c1,c2 be Function of Y, BOOLEAN holds ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '<' ((a2 'or' b2) 'or' c2)

    proof

      let a1,a2,b1,b2,c1,c2 be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) . z) = (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) . z) '&' (((a1 'or' b1) 'or' c1) . z)) by MARGREL1:def 20

      .= (((((a1 'imp' a2) '&' (b1 'imp' b2)) . z) '&' ((c1 'imp' c2) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by MARGREL1:def 20

      .= (((((a1 'imp' a2) . z) '&' ((b1 'imp' b2) . z)) '&' ((c1 'imp' c2) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by MARGREL1:def 20

      .= (((((( 'not' a1) 'or' a2) . z) '&' ((b1 'imp' b2) . z)) '&' ((c1 'imp' c2) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by BVFUNC_4: 8

      .= (((((( 'not' a1) 'or' a2) . z) '&' ((( 'not' b1) 'or' b2) . z)) '&' ((c1 'imp' c2) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by BVFUNC_4: 8

      .= (((((( 'not' a1) 'or' a2) . z) '&' ((( 'not' b1) 'or' b2) . z)) '&' ((( 'not' c1) 'or' c2) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by BVFUNC_4: 8

      .= (((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) 'or' b2) . z)) '&' ((( 'not' c1) 'or' c2) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) . z) 'or' (b2 . z))) '&' ((( 'not' c1) 'or' c2) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) . z) 'or' (b2 . z))) '&' ((( 'not' c1) . z) 'or' (c2 . z))) '&' (((a1 'or' b1) 'or' c1) . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) . z) 'or' (b2 . z))) '&' ((( 'not' c1) . z) 'or' (c2 . z))) '&' (((a1 'or' b1) . z) 'or' (c1 . z))) by BVFUNC_1:def 4

      .= (((((( 'not' a1) . z) 'or' (a2 . z)) '&' ((( 'not' b1) . z) 'or' (b2 . z))) '&' ((( 'not' c1) . z) 'or' (c2 . z))) '&' (((a1 . z) 'or' (b1 . z)) 'or' (c1 . z))) by BVFUNC_1:def 4

      .= ((((( 'not' (a1 . z)) 'or' (a2 . z)) '&' ((( 'not' b1) . z) 'or' (b2 . z))) '&' ((( 'not' c1) . z) 'or' (c2 . z))) '&' (((a1 . z) 'or' (b1 . z)) 'or' (c1 . z))) by MARGREL1:def 19

      .= ((((( 'not' (a1 . z)) 'or' (a2 . z)) '&' (( 'not' (b1 . z)) 'or' (b2 . z))) '&' ((( 'not' c1) . z) 'or' (c2 . z))) '&' (((a1 . z) 'or' (b1 . z)) 'or' (c1 . z))) by MARGREL1:def 19

      .= ((((( 'not' (a1 . z)) 'or' (a2 . z)) '&' (( 'not' (b1 . z)) 'or' (b2 . z))) '&' (( 'not' (c1 . z)) 'or' (c2 . z))) '&' (((a1 . z) 'or' (b1 . z)) 'or' (c1 . z))) by MARGREL1:def 19;

      assume

       A2: (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) . z) = TRUE ;

      now



         A3: (b2 . z) = TRUE or (b2 . z) = FALSE by XBOOLEAN:def 3;



         A4: (c2 . z) = TRUE or (c2 . z) = FALSE by XBOOLEAN:def 3;



         A5: (((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' ((a1 'or' b1) 'or' c1)) = ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' (a1 'or' b1)) 'or' (((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' c1)) by BVFUNC_1: 12

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' (a1 'or' b1)) 'or' ((( 'not' a1) '&' ( 'not' b1)) '&' (( 'not' c1) '&' c1))) by BVFUNC_1: 4

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' (a1 'or' b1)) 'or' ((( 'not' a1) '&' ( 'not' b1)) '&' ( O_el Y))) by BVFUNC_4: 5

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' (a1 'or' b1)) 'or' ( O_el Y)) by BVFUNC_1: 5

        .= (((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' (a1 'or' b1)) by BVFUNC_1: 9

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' a1) 'or' (((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' b1)) by BVFUNC_1: 12

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' a1) 'or' (((( 'not' a1) '&' ( 'not' c1)) '&' ( 'not' b1)) '&' b1)) by BVFUNC_1: 4

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' a1) 'or' ((( 'not' a1) '&' ( 'not' c1)) '&' (( 'not' b1) '&' b1))) by BVFUNC_1: 4

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' a1) 'or' ((( 'not' a1) '&' ( 'not' c1)) '&' ( O_el Y))) by BVFUNC_4: 5

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' a1) 'or' ( O_el Y)) by BVFUNC_1: 5

        .= (((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' a1) by BVFUNC_1: 9

        .= (((( 'not' b1) '&' ( 'not' c1)) '&' ( 'not' a1)) '&' a1) by BVFUNC_1: 4

        .= ((( 'not' b1) '&' ( 'not' c1)) '&' (( 'not' a1) '&' a1)) by BVFUNC_1: 4

        .= ((( 'not' b1) '&' ( 'not' c1)) '&' ( O_el Y)) by BVFUNC_4: 5

        .= ( O_el Y) by BVFUNC_1: 5;



         A6: ((a2 . z) 'or' (b2 . z)) = TRUE or ((a2 . z) 'or' (b2 . z)) = FALSE by XBOOLEAN:def 3;



         A7: (((a2 'or' b2) 'or' c2) . z) = (((a2 'or' b2) . z) 'or' (c2 . z)) by BVFUNC_1:def 4

        .= (((a2 . z) 'or' (b2 . z)) 'or' (c2 . z)) by BVFUNC_1:def 4;

        assume (((a2 'or' b2) 'or' c2) . z) <> TRUE ;



        then ((((( 'not' (a1 . z)) 'or' (a2 . z)) '&' (( 'not' (b1 . z)) 'or' (b2 . z))) '&' (( 'not' (c1 . z)) 'or' (c2 . z))) '&' (((a1 . z) 'or' (b1 . z)) 'or' (c1 . z))) = ((((( 'not' a1) . z) '&' ( 'not' (b1 . z))) '&' ( 'not' (c1 . z))) '&' (((a1 . z) 'or' (b1 . z)) 'or' (c1 . z))) by A7, A6, A4, A3, MARGREL1:def 19

        .= ((((( 'not' a1) . z) '&' (( 'not' b1) . z)) '&' ( 'not' (c1 . z))) '&' (((a1 . z) 'or' (b1 . z)) 'or' (c1 . z))) by MARGREL1:def 19

        .= ((((( 'not' a1) . z) '&' (( 'not' b1) . z)) '&' (( 'not' c1) . z)) '&' (((a1 . z) 'or' (b1 . z)) 'or' (c1 . z))) by MARGREL1:def 19

        .= ((((( 'not' a1) . z) '&' (( 'not' b1) . z)) '&' (( 'not' c1) . z)) '&' (((a1 'or' b1) . z) 'or' (c1 . z))) by BVFUNC_1:def 4

        .= ((((( 'not' a1) . z) '&' (( 'not' b1) . z)) '&' (( 'not' c1) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by BVFUNC_1:def 4

        .= ((((( 'not' a1) '&' ( 'not' b1)) . z) '&' (( 'not' c1) . z)) '&' (((a1 'or' b1) 'or' c1) . z)) by MARGREL1:def 20

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) . z) '&' (((a1 'or' b1) 'or' c1) . z)) by MARGREL1:def 20

        .= ((((( 'not' a1) '&' ( 'not' b1)) '&' ( 'not' c1)) '&' ((a1 'or' b1) 'or' c1)) . z) by MARGREL1:def 20;

        hence contradiction by A2, A1, A5, BVFUNC_1:def 10;

      end;

      hence thesis;

    end;



     Lm3: for a1,a2,b1,b2 be Function of Y, BOOLEAN holds ((((a1 'imp' b1) '&' (a2 'imp' b2)) '&' (a1 'or' a2)) '&' ( 'not' (b1 '&' b2))) '<' ((((b1 'imp' a1) '&' (b2 'imp' a2)) '&' (b1 'or' b2)) '&' ( 'not' (a1 '&' a2)))

    proof

      let a1,a2,b1,b2 be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((((a1 'imp' b1) '&' (a2 'imp' b2)) '&' (a1 'or' a2)) '&' ( 'not' (b1 '&' b2))) . z) = (((((( 'not' a1) 'or' b1) '&' (a2 'imp' b2)) '&' (a1 'or' a2)) '&' ( 'not' (b1 '&' b2))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a1) 'or' b1) '&' (( 'not' a2) 'or' b2)) '&' (a1 'or' a2)) '&' ( 'not' (b1 '&' b2))) . z) by BVFUNC_4: 8

      .= (((((( 'not' a1) 'or' b1) '&' (( 'not' a2) 'or' b2)) '&' (a1 'or' a2)) . z) '&' (( 'not' (b1 '&' b2)) . z)) by MARGREL1:def 20

      .= (((((( 'not' a1) 'or' b1) '&' (( 'not' a2) 'or' b2)) . z) '&' ((a1 'or' a2) . z)) '&' (( 'not' (b1 '&' b2)) . z)) by MARGREL1:def 20

      .= (((((( 'not' a1) 'or' b1) '&' (( 'not' a2) 'or' b2)) . z) '&' ((a1 'or' a2) . z)) '&' ((( 'not' b1) 'or' ( 'not' b2)) . z)) by BVFUNC_1: 14

      .= (((((( 'not' a1) 'or' b1) . z) '&' ((( 'not' a2) 'or' b2) . z)) '&' ((a1 'or' a2) . z)) '&' ((( 'not' b1) 'or' ( 'not' b2)) . z)) by MARGREL1:def 20

      .= (((((( 'not' a1) . z) 'or' (b1 . z)) '&' ((( 'not' a2) 'or' b2) . z)) '&' ((a1 'or' a2) . z)) '&' ((( 'not' b1) 'or' ( 'not' b2)) . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a1) . z) 'or' (b1 . z)) '&' ((( 'not' a2) . z) 'or' (b2 . z))) '&' ((a1 'or' a2) . z)) '&' ((( 'not' b1) 'or' ( 'not' b2)) . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a1) . z) 'or' (b1 . z)) '&' ((( 'not' a2) . z) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' ((( 'not' b1) 'or' ( 'not' b2)) . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a1) . z) 'or' (b1 . z)) '&' ((( 'not' a2) . z) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' ((( 'not' b1) . z) 'or' (( 'not' b2) . z))) by BVFUNC_1:def 4

      .= ((((( 'not' (a1 . z)) 'or' (b1 . z)) '&' ((( 'not' a2) . z) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' ((( 'not' b1) . z) 'or' (( 'not' b2) . z))) by MARGREL1:def 19

      .= ((((( 'not' (a1 . z)) 'or' (b1 . z)) '&' (( 'not' (a2 . z)) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' ((( 'not' b1) . z) 'or' (( 'not' b2) . z))) by MARGREL1:def 19

      .= ((((( 'not' (a1 . z)) 'or' (b1 . z)) '&' (( 'not' (a2 . z)) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' (( 'not' (b1 . z)) 'or' (( 'not' b2) . z))) by MARGREL1:def 19

      .= ((((( 'not' (a1 . z)) 'or' (b1 . z)) '&' (( 'not' (a2 . z)) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' (( 'not' (b1 . z)) 'or' ( 'not' (b2 . z)))) by MARGREL1:def 19;

      assume

       A2: (((((a1 'imp' b1) '&' (a2 'imp' b2)) '&' (a1 'or' a2)) '&' ( 'not' (b1 '&' b2))) . z) = TRUE ;

      now



         A3: (((((b1 'imp' a1) '&' (b2 'imp' a2)) '&' (b1 'or' b2)) '&' ( 'not' (a1 '&' a2))) . z) = (((((( 'not' b1) 'or' a1) '&' (b2 'imp' a2)) '&' (b1 'or' b2)) '&' ( 'not' (a1 '&' a2))) . z) by BVFUNC_4: 8

        .= (((((( 'not' b1) 'or' a1) '&' (( 'not' b2) 'or' a2)) '&' (b1 'or' b2)) '&' ( 'not' (a1 '&' a2))) . z) by BVFUNC_4: 8

        .= (((((( 'not' b1) 'or' a1) '&' (( 'not' b2) 'or' a2)) '&' (b1 'or' b2)) . z) '&' (( 'not' (a1 '&' a2)) . z)) by MARGREL1:def 20

        .= (((((( 'not' b1) 'or' a1) '&' (( 'not' b2) 'or' a2)) . z) '&' ((b1 'or' b2) . z)) '&' (( 'not' (a1 '&' a2)) . z)) by MARGREL1:def 20

        .= (((((( 'not' b1) 'or' a1) . z) '&' ((( 'not' b2) 'or' a2) . z)) '&' ((b1 'or' b2) . z)) '&' (( 'not' (a1 '&' a2)) . z)) by MARGREL1:def 20

        .= (((((( 'not' b1) 'or' a1) . z) '&' ((( 'not' b2) 'or' a2) . z)) '&' ((b1 'or' b2) . z)) '&' ((( 'not' a1) 'or' ( 'not' a2)) . z)) by BVFUNC_1: 14

        .= (((((( 'not' b1) . z) 'or' (a1 . z)) '&' ((( 'not' b2) 'or' a2) . z)) '&' ((b1 'or' b2) . z)) '&' ((( 'not' a1) 'or' ( 'not' a2)) . z)) by BVFUNC_1:def 4

        .= (((((( 'not' b1) . z) 'or' (a1 . z)) '&' ((( 'not' b2) . z) 'or' (a2 . z))) '&' ((b1 'or' b2) . z)) '&' ((( 'not' a1) 'or' ( 'not' a2)) . z)) by BVFUNC_1:def 4

        .= (((((( 'not' b1) . z) 'or' (a1 . z)) '&' ((( 'not' b2) . z) 'or' (a2 . z))) '&' ((b1 . z) 'or' (b2 . z))) '&' ((( 'not' a1) 'or' ( 'not' a2)) . z)) by BVFUNC_1:def 4

        .= (((((( 'not' b1) . z) 'or' (a1 . z)) '&' ((( 'not' b2) . z) 'or' (a2 . z))) '&' ((b1 . z) 'or' (b2 . z))) '&' ((( 'not' a1) . z) 'or' (( 'not' a2) . z))) by BVFUNC_1:def 4;

        assume (((((b1 'imp' a1) '&' (b2 'imp' a2)) '&' (b1 'or' b2)) '&' ( 'not' (a1 '&' a2))) . z) <> TRUE ;

        then

         A4: (((((( 'not' b1) . z) 'or' (a1 . z)) '&' ((( 'not' b2) . z) 'or' (a2 . z))) '&' ((b1 . z) 'or' (b2 . z))) '&' ((( 'not' a1) . z) 'or' (( 'not' a2) . z))) = FALSE by A3, XBOOLEAN:def 3;

        now

          per cases by A4, MARGREL1: 12;

            case

             A5: ((((( 'not' b1) . z) 'or' (a1 . z)) '&' ((( 'not' b2) . z) 'or' (a2 . z))) '&' ((b1 . z) 'or' (b2 . z))) = FALSE ;

            now

              per cases by A5, MARGREL1: 12;

                case

                 A6: (((( 'not' b1) . z) 'or' (a1 . z)) '&' ((( 'not' b2) . z) 'or' (a2 . z))) = FALSE ;

                now

                  per cases by A6, MARGREL1: 12;

                    case

                     A7: ((( 'not' b1) . z) 'or' (a1 . z)) = FALSE ;



                     A8: (a1 . z) = TRUE or (a1 . z) = FALSE by XBOOLEAN:def 3;

                    then ( 'not' (b1 . z)) = FALSE by A7, MARGREL1:def 19;



                    then ((((( 'not' (a1 . z)) 'or' (b1 . z)) '&' (( 'not' (a2 . z)) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' (( 'not' (b1 . z)) 'or' ( 'not' (b2 . z)))) = ((((a2 . z) '&' ( 'not' (a2 . z))) 'or' ((a2 . z) '&' (b2 . z))) '&' ( 'not' (b2 . z))) by A7, A8, XBOOLEAN: 8

                    .= (( FALSE 'or' ((a2 . z) '&' (b2 . z))) '&' ( 'not' (b2 . z))) by XBOOLEAN: 138

                    .= (((a2 . z) '&' (b2 . z)) '&' ( 'not' (b2 . z)))

                    .= ((a2 . z) '&' ((b2 . z) '&' ( 'not' (b2 . z))))

                    .= ( FALSE '&' (a2 . z)) by XBOOLEAN: 138

                    .= FALSE ;

                    hence thesis by A2, A1;

                  end;

                    case

                     A9: ((( 'not' b2) . z) 'or' (a2 . z)) = FALSE ;



                     A10: (a2 . z) = TRUE or (a2 . z) = FALSE by XBOOLEAN:def 3;

                    then ( 'not' (b2 . z)) = FALSE by A9, MARGREL1:def 19;



                    then ((((( 'not' (a1 . z)) 'or' (b1 . z)) '&' (( 'not' (a2 . z)) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' (( 'not' (b1 . z)) 'or' ( 'not' (b2 . z)))) = ((((a1 . z) '&' ( 'not' (a1 . z))) 'or' ((a1 . z) '&' (b1 . z))) '&' ( 'not' (b1 . z))) by A9, A10, XBOOLEAN: 8

                    .= (( FALSE 'or' ((a1 . z) '&' (b1 . z))) '&' ( 'not' (b1 . z))) by XBOOLEAN: 138

                    .= (((a1 . z) '&' (b1 . z)) '&' ( 'not' (b1 . z)))

                    .= ((a1 . z) '&' ((b1 . z) '&' ( 'not' (b1 . z))))

                    .= ( FALSE '&' (a1 . z)) by XBOOLEAN: 138

                    .= FALSE ;

                    hence thesis by A2, A1;

                  end;

                end;

                hence thesis;

              end;

                case

                 A11: ((b1 . z) 'or' (b2 . z)) = FALSE ;

                reconsider a2z = (a2 . z) as boolean object;

                reconsider a1z = (a1 . z) as boolean object;

                (b1 . z) = TRUE or (b1 . z) = FALSE by XBOOLEAN:def 3;



                then ((((( 'not' (a1 . z)) 'or' (b1 . z)) '&' (( 'not' (a2 . z)) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' (( 'not' (b1 . z)) 'or' ( 'not' (b2 . z)))) = (( 'not' (a1 . z)) '&' (( 'not' (a2 . z)) '&' ((a1 . z) 'or' (a2 . z)))) by A11

                .= (( 'not' (a1 . z)) '&' ((( 'not' (a2 . z)) '&' (a1 . z)) 'or' (( 'not' a2z) '&' a2z))) by XBOOLEAN: 8

                .= (( 'not' (a1 . z)) '&' ((( 'not' (a2 . z)) '&' (a1 . z)) 'or' FALSE )) by XBOOLEAN: 138

                .= (( 'not' (a1 . z)) '&' ((a1 . z) '&' ( 'not' (a2 . z))))

                .= ((( 'not' a1z) '&' a1z) '&' ( 'not' (a2 . z)))

                .= ( FALSE '&' ( 'not' (a2 . z))) by XBOOLEAN: 138

                .= FALSE ;

                hence thesis by A2, A1;

              end;

            end;

            hence thesis;

          end;

            case

             A12: ((( 'not' a1) . z) 'or' (( 'not' a2) . z)) = FALSE ;

            (( 'not' a2) . z) = TRUE or (( 'not' a2) . z) = FALSE by XBOOLEAN:def 3;

            then ( 'not' (a1 . z)) = FALSE & ( 'not' (a2 . z)) = FALSE by A12, MARGREL1:def 19;



            then ((((( 'not' (a1 . z)) 'or' (b1 . z)) '&' (( 'not' (a2 . z)) 'or' (b2 . z))) '&' ((a1 . z) 'or' (a2 . z))) '&' (( 'not' (b1 . z)) 'or' ( 'not' (b2 . z)))) = ((b1 . z) '&' ((b2 . z) '&' (( 'not' (b1 . z)) 'or' ( 'not' (b2 . z)))))

            .= ((b1 . z) '&' (((b2 . z) '&' ( 'not' (b1 . z))) 'or' ((b2 . z) '&' ( 'not' (b2 . z))))) by XBOOLEAN: 8

            .= ((b1 . z) '&' (((b2 . z) '&' ( 'not' (b1 . z))) 'or' FALSE )) by XBOOLEAN: 138

            .= ((b1 . z) '&' (( 'not' (b1 . z)) '&' (b2 . z)))

            .= (((b1 . z) '&' ( 'not' (b1 . z))) '&' (b2 . z))

            .= ( FALSE '&' (b2 . z)) by XBOOLEAN: 138

            .= FALSE ;

            hence thesis by A2, A1;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:148

    for a1,a2,b1,b2 be Function of Y, BOOLEAN holds ((((a1 'imp' b1) '&' (a2 'imp' b2)) '&' (a1 'or' a2)) '&' ( 'not' (b1 '&' b2))) = ((((b1 'imp' a1) '&' (b2 'imp' a2)) '&' (b1 'or' b2)) '&' ( 'not' (a1 '&' a2)))

    proof

      let a1,a2,b1,b2 be Function of Y, BOOLEAN ;

      ((((a1 'imp' b1) '&' (a2 'imp' b2)) '&' (a1 'or' a2)) '&' ( 'not' (b1 '&' b2))) '<' ((((b1 'imp' a1) '&' (b2 'imp' a2)) '&' (b1 'or' b2)) '&' ( 'not' (a1 '&' a2))) & ((((b1 'imp' a1) '&' (b2 'imp' a2)) '&' (b1 'or' b2)) '&' ( 'not' (a1 '&' a2))) '<' ((((a1 'imp' b1) '&' (a2 'imp' b2)) '&' (a1 'or' a2)) '&' ( 'not' (b1 '&' b2))) by Lm3;

      hence thesis by BVFUNC_1: 15;

    end;

    theorem :: BVFUNC_6:149

    for a,b,c,d be Function of Y, BOOLEAN holds ((a 'or' b) '&' (c 'or' d)) = ((((a '&' c) 'or' (a '&' d)) 'or' (b '&' c)) 'or' (b '&' d))

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      ((a 'or' b) '&' (c 'or' d)) = (((a 'or' b) '&' c) 'or' ((a 'or' b) '&' d)) by BVFUNC_1: 12

      .= (((a '&' c) 'or' (b '&' c)) 'or' ((a 'or' b) '&' d)) by BVFUNC_1: 12

      .= (((a '&' c) 'or' (b '&' c)) 'or' ((a '&' d) 'or' (b '&' d))) by BVFUNC_1: 12

      .= ((((a '&' c) 'or' (b '&' c)) 'or' (a '&' d)) 'or' (b '&' d)) by BVFUNC_1: 8

      .= ((((a '&' c) 'or' (a '&' d)) 'or' (b '&' c)) 'or' (b '&' d)) by BVFUNC_1: 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:150

    for a1,a2,b1,b2,b3 be Function of Y, BOOLEAN holds ((a1 '&' a2) 'or' ((b1 '&' b2) '&' b3)) = ((((((a1 'or' b1) '&' (a1 'or' b2)) '&' (a1 'or' b3)) '&' (a2 'or' b1)) '&' (a2 'or' b2)) '&' (a2 'or' b3))

    proof

      let a1,a2,b1,b2,b3 be Function of Y, BOOLEAN ;

      ((((((a1 'or' b1) '&' (a1 'or' b2)) '&' (a1 'or' b3)) '&' (a2 'or' b1)) '&' (a2 'or' b2)) '&' (a2 'or' b3)) = (((((a1 'or' (b1 '&' b2)) '&' (a1 'or' b3)) '&' (a2 'or' b1)) '&' (a2 'or' b2)) '&' (a2 'or' b3)) by BVFUNC_1: 11

      .= ((((a1 'or' ((b1 '&' b2) '&' b3)) '&' (a2 'or' b1)) '&' (a2 'or' b2)) '&' (a2 'or' b3)) by BVFUNC_1: 11

      .= (((a1 'or' ((b1 '&' b2) '&' b3)) '&' ((a2 'or' b1) '&' (a2 'or' b2))) '&' (a2 'or' b3)) by BVFUNC_1: 4

      .= ((a1 'or' ((b1 '&' b2) '&' b3)) '&' (((a2 'or' b1) '&' (a2 'or' b2)) '&' (a2 'or' b3))) by BVFUNC_1: 4

      .= ((a1 'or' ((b1 '&' b2) '&' b3)) '&' ((a2 'or' (b1 '&' b2)) '&' (a2 'or' b3))) by BVFUNC_1: 11

      .= ((a1 'or' ((b1 '&' b2) '&' b3)) '&' (a2 'or' ((b1 '&' b2) '&' b3))) by BVFUNC_1: 11

      .= ((a1 '&' a2) 'or' ((b1 '&' b2) '&' b3)) by BVFUNC_1: 11;

      hence thesis;

    end;

    theorem :: BVFUNC_6:151

    for a,b,c,d be Function of Y, BOOLEAN holds (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) = (((a 'imp' ((b '&' c) '&' d)) '&' (b 'imp' (c '&' d))) '&' (c 'imp' d))

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;



       A1: (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) = ((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) '&' (c 'imp' d)) by Th12

      .= (((a 'imp' b) '&' (b 'imp' c)) '&' ((a 'imp' c) '&' (c 'imp' d))) by BVFUNC_1: 4

      .= (((a 'imp' b) '&' (b 'imp' c)) '&' (((a 'imp' c) '&' (c 'imp' d)) '&' (a 'imp' d))) by Th12

      .= ((((a 'imp' b) '&' (b 'imp' c)) '&' ((a 'imp' c) '&' (c 'imp' d))) '&' (a 'imp' d)) by BVFUNC_1: 4

      .= (((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) '&' (a 'imp' c)) '&' (a 'imp' d)) by BVFUNC_1: 4

      .= ((((a 'imp' b) '&' ((b 'imp' c) '&' (c 'imp' d))) '&' (a 'imp' c)) '&' (a 'imp' d)) by BVFUNC_1: 4

      .= ((((a 'imp' b) '&' (((b 'imp' c) '&' (c 'imp' d)) '&' (b 'imp' d))) '&' (a 'imp' c)) '&' (a 'imp' d)) by Th12

      .= (((((a 'imp' b) '&' ((b 'imp' c) '&' (c 'imp' d))) '&' (b 'imp' d)) '&' (a 'imp' c)) '&' (a 'imp' d)) by BVFUNC_1: 4

      .= ((((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) '&' (b 'imp' d)) '&' (a 'imp' c)) '&' (a 'imp' d)) by BVFUNC_1: 4

      .= ((((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) '&' (a 'imp' c)) '&' (b 'imp' d)) '&' (a 'imp' d)) by BVFUNC_1: 4

      .= ((((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) '&' (a 'imp' c)) '&' (a 'imp' d)) '&' (b 'imp' d)) by BVFUNC_1: 4;

      (((a 'imp' ((b '&' c) '&' d)) '&' (b 'imp' (c '&' d))) '&' (c 'imp' d)) = (((( 'not' a) 'or' ((b '&' c) '&' d)) '&' (b 'imp' (c '&' d))) '&' (c 'imp' d)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' ((b '&' c) '&' d)) '&' (( 'not' b) 'or' (c '&' d))) '&' (c 'imp' d)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' ((b '&' c) '&' d)) '&' (( 'not' b) 'or' (c '&' d))) '&' (( 'not' c) 'or' d)) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' (c '&' d))) '&' (( 'not' c) 'or' d)) by BVFUNC_5: 39

      .= (((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' ((( 'not' b) 'or' c) '&' (( 'not' b) 'or' d))) '&' (( 'not' c) 'or' d)) by BVFUNC_1: 11

      .= ((((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' c)) '&' (( 'not' b) 'or' d)) '&' (( 'not' c) 'or' d)) by BVFUNC_1: 4

      .= ((((((( 'not' a) 'or' b) '&' (( 'not' a) 'or' c)) '&' (( 'not' b) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' d)) '&' (( 'not' c) 'or' d)) by BVFUNC_1: 4

      .= ((((((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' d)) '&' (( 'not' c) 'or' d)) by BVFUNC_1: 4

      .= ((((((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' c) 'or' d)) '&' (( 'not' b) 'or' d)) by BVFUNC_1: 4

      .= ((((((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' a) 'or' c)) '&' (( 'not' c) 'or' d)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' d)) by BVFUNC_1: 4

      .= ((((((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' d)) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' d)) by BVFUNC_1: 4

      .= ((((((a 'imp' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' d)) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' d)) by BVFUNC_4: 8

      .= ((((((a 'imp' b) '&' (b 'imp' c)) '&' (( 'not' c) 'or' d)) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' d)) by BVFUNC_4: 8

      .= ((((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) '&' (( 'not' a) 'or' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' d)) by BVFUNC_4: 8

      .= ((((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) '&' (a 'imp' c)) '&' (( 'not' a) 'or' d)) '&' (( 'not' b) 'or' d)) by BVFUNC_4: 8

      .= ((((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) '&' (a 'imp' c)) '&' (a 'imp' d)) '&' (( 'not' b) 'or' d)) by BVFUNC_4: 8

      .= ((((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d)) '&' (a 'imp' c)) '&' (a 'imp' d)) '&' (b 'imp' d)) by BVFUNC_4: 8;

      hence thesis by A1;

    end;

    theorem :: BVFUNC_6:152

    for a,b,c,d be Function of Y, BOOLEAN holds (((a 'imp' c) '&' (b 'imp' d)) '&' (a 'or' b)) '<' (c 'or' d)

    proof

      let a,b,c,d be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: ((((a 'imp' c) '&' (b 'imp' d)) '&' (a 'or' b)) . z) = ((((a 'imp' c) '&' (b 'imp' d)) . z) '&' ((a 'or' b) . z)) by MARGREL1:def 20

      .= ((((a 'imp' c) . z) '&' ((b 'imp' d) . z)) '&' ((a 'or' b) . z)) by MARGREL1:def 20

      .= ((((( 'not' a) 'or' c) . z) '&' ((b 'imp' d) . z)) '&' ((a 'or' b) . z)) by BVFUNC_4: 8

      .= ((((( 'not' a) 'or' c) . z) '&' ((( 'not' b) 'or' d) . z)) '&' ((a 'or' b) . z)) by BVFUNC_4: 8

      .= ((((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' b) 'or' d) . z)) '&' ((a 'or' b) . z)) by BVFUNC_1:def 4

      .= ((((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' b) . z) 'or' (d . z))) '&' ((a 'or' b) . z)) by BVFUNC_1:def 4

      .= ((((( 'not' a) . z) 'or' (c . z)) '&' ((( 'not' b) . z) 'or' (d . z))) '&' ((a . z) 'or' (b . z))) by BVFUNC_1:def 4

      .= (((( 'not' (a . z)) 'or' (c . z)) '&' ((( 'not' b) . z) 'or' (d . z))) '&' ((a . z) 'or' (b . z))) by MARGREL1:def 19

      .= (((( 'not' (a . z)) 'or' (c . z)) '&' (( 'not' (b . z)) 'or' (d . z))) '&' ((a . z) 'or' (b . z))) by MARGREL1:def 19;

      reconsider bz = (b . z) as boolean object;

      reconsider az = (a . z) as boolean object;

      assume

       A2: ((((a 'imp' c) '&' (b 'imp' d)) '&' (a 'or' b)) . z) = TRUE ;

      now

        assume ((c 'or' d) . z) <> TRUE ;

        then ((c 'or' d) . z) = FALSE by XBOOLEAN:def 3;

        then

         A3: ((c . z) 'or' (d . z)) = FALSE by BVFUNC_1:def 4;

        (d . z) = TRUE or (d . z) = FALSE by XBOOLEAN:def 3;



        then (((( 'not' (a . z)) 'or' (c . z)) '&' (( 'not' (b . z)) 'or' (d . z))) '&' ((a . z) 'or' (b . z))) = (( 'not' (a . z)) '&' (( 'not' (b . z)) '&' ((b . z) 'or' (a . z)))) by A3

        .= (( 'not' (a . z)) '&' ((( 'not' bz) '&' bz) 'or' (( 'not' (b . z)) '&' (a . z)))) by XBOOLEAN: 8

        .= (( 'not' (a . z)) '&' ( FALSE 'or' (( 'not' (b . z)) '&' (a . z)))) by XBOOLEAN: 138

        .= (( 'not' (a . z)) '&' ((a . z) '&' ( 'not' (b . z))))

        .= ((( 'not' az) '&' az) '&' ( 'not' (b . z)))

        .= ( FALSE '&' ( 'not' (b . z))) by XBOOLEAN: 138

        .= FALSE ;

        hence thesis by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:153

    for a,b,c be Function of Y, BOOLEAN holds ((((a '&' b) 'imp' ( 'not' c)) '&' a) '&' c) '<' ( 'not' b)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((((a '&' b) 'imp' ( 'not' c)) '&' a) '&' c) . z) = (((((a '&' b) 'imp' ( 'not' c)) '&' a) . z) '&' (c . z)) by MARGREL1:def 20

      .= (((((a '&' b) 'imp' ( 'not' c)) . z) '&' (a . z)) '&' (c . z)) by MARGREL1:def 20

      .= ((((( 'not' (a '&' b)) 'or' ( 'not' c)) . z) '&' (a . z)) '&' (c . z)) by BVFUNC_4: 8

      .= (((((( 'not' a) 'or' ( 'not' b)) 'or' ( 'not' c)) . z) '&' (a . z)) '&' (c . z)) by BVFUNC_1: 14

      .= (((((( 'not' a) 'or' ( 'not' b)) . z) 'or' (( 'not' c) . z)) '&' (a . z)) '&' (c . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a) . z) 'or' (( 'not' b) . z)) 'or' (( 'not' c) . z)) '&' (a . z)) '&' (c . z)) by BVFUNC_1:def 4;

      reconsider cz = (c . z) as boolean object;

      assume

       A2: (((((a '&' b) 'imp' ( 'not' c)) '&' a) '&' c) . z) = TRUE ;

      now

        assume (( 'not' b) . z) <> TRUE ;

        then (( 'not' b) . z) = FALSE by XBOOLEAN:def 3;



        then (((((( 'not' a) . z) 'or' (( 'not' b) . z)) 'or' (( 'not' c) . z)) '&' (a . z)) '&' (c . z)) = (((( 'not' (a . z)) 'or' (( 'not' c) . z)) '&' (a . z)) '&' (c . z)) by MARGREL1:def 19

        .= (((a . z) '&' (( 'not' (a . z)) 'or' ( 'not' (c . z)))) '&' (c . z)) by MARGREL1:def 19

        .= ((((a . z) '&' ( 'not' (a . z))) 'or' ((a . z) '&' ( 'not' (c . z)))) '&' (c . z)) by XBOOLEAN: 8

        .= (( FALSE 'or' ((a . z) '&' ( 'not' (c . z)))) '&' (c . z)) by XBOOLEAN: 138

        .= (((a . z) '&' ( 'not' (c . z))) '&' (c . z))

        .= ((a . z) '&' (( 'not' cz) '&' cz))

        .= ( FALSE '&' (a . z)) by XBOOLEAN: 138

        .= FALSE ;

        hence thesis by A2, A1;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:154

    for a1,a2,a3,b1,b2,b3 be Function of Y, BOOLEAN holds (((a1 '&' a2) '&' a3) 'imp' ((b1 'or' b2) 'or' b3)) = (((( 'not' b1) '&' ( 'not' b2)) '&' a3) 'imp' ((( 'not' a1) 'or' ( 'not' a2)) 'or' b3))

    proof

      let a1,a2,a3,b1,b2,b3 be Function of Y, BOOLEAN ;

      (((( 'not' b1) '&' ( 'not' b2)) '&' a3) 'imp' ((( 'not' a1) 'or' ( 'not' a2)) 'or' b3)) = (( 'not' ((( 'not' b1) '&' ( 'not' b2)) '&' a3)) 'or' ((( 'not' a1) 'or' ( 'not' a2)) 'or' b3)) by BVFUNC_4: 8

      .= (((( 'not' ( 'not' b1)) 'or' ( 'not' ( 'not' b2))) 'or' ( 'not' a3)) 'or' ((( 'not' a1) 'or' ( 'not' a2)) 'or' b3)) by BVFUNC_5: 37

      .= ((((b1 'or' b2) 'or' ( 'not' a3)) 'or' (( 'not' a1) 'or' ( 'not' a2))) 'or' b3) by BVFUNC_1: 8

      .= (((b1 'or' b2) 'or' ((( 'not' a1) 'or' ( 'not' a2)) 'or' ( 'not' a3))) 'or' b3) by BVFUNC_1: 8

      .= (((( 'not' a1) 'or' ( 'not' a2)) 'or' ( 'not' a3)) 'or' ((b1 'or' b2) 'or' b3)) by BVFUNC_1: 8

      .= (( 'not' ((a1 '&' a2) '&' a3)) 'or' ((b1 'or' b2) 'or' b3)) by BVFUNC_5: 37

      .= (((a1 '&' a2) '&' a3) 'imp' ((b1 'or' b2) 'or' b3)) by BVFUNC_4: 8;

      hence thesis;

    end;

    theorem :: BVFUNC_6:155

    for a,b,c be Function of Y, BOOLEAN holds (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) = (((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      for z be Element of Y st ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) . z) = TRUE holds ((((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c))) . z) = TRUE

      proof

        let z be Element of Y;



         A1: ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) . z) = ((((a 'imp' b) '&' (b 'imp' c)) . z) '&' ((c 'imp' a) . z)) by MARGREL1:def 20

        .= ((((a 'imp' b) . z) '&' ((b 'imp' c) . z)) '&' ((c 'imp' a) . z)) by MARGREL1:def 20

        .= ((((( 'not' a) 'or' b) . z) '&' ((b 'imp' c) . z)) '&' ((c 'imp' a) . z)) by BVFUNC_4: 8

        .= ((((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) '&' ((c 'imp' a) . z)) by BVFUNC_4: 8

        .= ((((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) '&' ((( 'not' c) 'or' a) . z)) by BVFUNC_4: 8

        .= ((((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' c) . z)) '&' ((( 'not' c) 'or' a) . z)) by BVFUNC_1:def 4

        .= ((((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) '&' ((( 'not' c) 'or' a) . z)) by BVFUNC_1:def 4

        .= ((((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) '&' ((( 'not' c) . z) 'or' (a . z))) by BVFUNC_1:def 4

        .= (((( 'not' (a . z)) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) '&' ((( 'not' c) . z) 'or' (a . z))) by MARGREL1:def 19

        .= (((( 'not' (a . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' (c . z))) '&' ((( 'not' c) . z) 'or' (a . z))) by MARGREL1:def 19

        .= (((( 'not' (a . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' (c . z))) '&' (( 'not' (c . z)) 'or' (a . z))) by MARGREL1:def 19;

        assume

         A2: ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) . z) = TRUE ;

        now



           A3: ((( 'not' (a . z)) '&' ( 'not' (b . z))) '&' ( 'not' (c . z))) = TRUE or ((( 'not' (a . z)) '&' ( 'not' (b . z))) '&' ( 'not' (c . z))) = FALSE by XBOOLEAN:def 3;

          assume

           A4: ((((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c))) . z) <> TRUE ;



           A5: ((((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c))) . z) = ((((a '&' b) '&' c) . z) 'or' (((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c)) . z)) by BVFUNC_1:def 4

          .= ((((a '&' b) . z) '&' (c . z)) 'or' (((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c)) . z)) by MARGREL1:def 20

          .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' (((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c)) . z)) by MARGREL1:def 20

          .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' (((( 'not' a) '&' ( 'not' b)) . z) '&' (( 'not' c) . z))) by MARGREL1:def 20

          .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' (((( 'not' a) . z) '&' (( 'not' b) . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

          .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' ((( 'not' (a . z)) '&' (( 'not' b) . z)) '&' (( 'not' c) . z))) by MARGREL1:def 19

          .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' ((( 'not' (a . z)) '&' ( 'not' (b . z))) '&' (( 'not' c) . z))) by MARGREL1:def 19

          .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' ((( 'not' (a . z)) '&' ( 'not' (b . z))) '&' ( 'not' (c . z)))) by MARGREL1:def 19;



           A6: (((a . z) '&' (b . z)) '&' (c . z)) = TRUE or (((a . z) '&' (b . z)) '&' (c . z)) = FALSE by XBOOLEAN:def 3;

          now

            per cases by A4, A5, A6, MARGREL1: 12;

              case

               A7: ((a . z) '&' (b . z)) = FALSE ;

              now

                per cases by A7, MARGREL1: 12;

                  case

                   A8: (a . z) = FALSE ;

                  now

                    per cases by A4, A5, A3, A8, MARGREL1: 12;

                      case ( 'not' (b . z)) = FALSE ;

                      hence thesis by A2, A1, A8, XBOOLEAN: 138;

                    end;

                      case ( 'not' (c . z)) = FALSE ;

                      hence thesis by A2, A1, A8;

                    end;

                  end;

                  hence thesis;

                end;

                  case

                   A9: (b . z) = FALSE ;

                  now

                    per cases by A4, A5, A3, A9, MARGREL1: 12;

                      case ( 'not' (a . z)) = FALSE ;

                      hence thesis by A2, A1, A9;

                    end;

                      case ( 'not' (c . z)) = FALSE ;

                      hence thesis by A2, A1, A9, XBOOLEAN: 138;

                    end;

                  end;

                  hence thesis;

                end;

              end;

              hence thesis;

            end;

              case

               A10: (c . z) = FALSE ;

              now

                per cases by A4, A5, A3, A10, MARGREL1: 12;

                  case ( 'not' (a . z)) = FALSE ;

                  hence thesis by A2, A1, A10, XBOOLEAN: 138;

                end;

                  case ( 'not' (b . z)) = FALSE ;

                  hence thesis by A2, A1, A10;

                end;

              end;

              hence thesis;

            end;

          end;

          hence thesis;

        end;

        hence thesis;

      end;

      then

       A11: (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '<' (((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c)));

      for z be Element of Y st ((((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c))) . z) = TRUE holds ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) . z) = TRUE

      proof

        let z be Element of Y;



         A12: ((((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c))) . z) = ((((a '&' b) '&' c) . z) 'or' (((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= ((((a '&' b) . z) '&' (c . z)) 'or' (((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c)) . z)) by MARGREL1:def 20

        .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' (((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c)) . z)) by MARGREL1:def 20

        .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' (((( 'not' a) '&' ( 'not' b)) . z) '&' (( 'not' c) . z))) by MARGREL1:def 20

        .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' (((( 'not' a) . z) '&' (( 'not' b) . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

        .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' ((( 'not' (a . z)) '&' (( 'not' b) . z)) '&' (( 'not' c) . z))) by MARGREL1:def 19

        .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' ((( 'not' (a . z)) '&' ( 'not' (b . z))) '&' (( 'not' c) . z))) by MARGREL1:def 19

        .= ((((a . z) '&' (b . z)) '&' (c . z)) 'or' ((( 'not' (a . z)) '&' ( 'not' (b . z))) '&' ( 'not' (c . z)))) by MARGREL1:def 19;

        assume

         A13: ((((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c))) . z) = TRUE ;

        now



           A14: ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) . z) = ((((a 'imp' b) '&' (b 'imp' c)) . z) '&' ((c 'imp' a) . z)) by MARGREL1:def 20

          .= ((((a 'imp' b) . z) '&' ((b 'imp' c) . z)) '&' ((c 'imp' a) . z)) by MARGREL1:def 20

          .= ((((( 'not' a) 'or' b) . z) '&' ((b 'imp' c) . z)) '&' ((c 'imp' a) . z)) by BVFUNC_4: 8

          .= ((((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) '&' ((c 'imp' a) . z)) by BVFUNC_4: 8

          .= ((((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) '&' ((( 'not' c) 'or' a) . z)) by BVFUNC_4: 8

          .= ((((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' c) . z)) '&' ((( 'not' c) 'or' a) . z)) by BVFUNC_1:def 4

          .= ((((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) '&' ((( 'not' c) 'or' a) . z)) by BVFUNC_1:def 4

          .= ((((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) '&' ((( 'not' c) . z) 'or' (a . z))) by BVFUNC_1:def 4

          .= (((( 'not' (a . z)) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) '&' ((( 'not' c) . z) 'or' (a . z))) by MARGREL1:def 19

          .= (((( 'not' (a . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' (c . z))) '&' ((( 'not' c) . z) 'or' (a . z))) by MARGREL1:def 19

          .= (((( 'not' (a . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' (c . z))) '&' (( 'not' (c . z)) 'or' (a . z))) by MARGREL1:def 19;

          assume ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) . z) <> TRUE ;

          then

           A15: (((( 'not' (a . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' (c . z))) '&' (( 'not' (c . z)) 'or' (a . z))) = FALSE by A14, XBOOLEAN:def 3;

          now

            per cases by A15, MARGREL1: 12;

              case

               A16: ((( 'not' (a . z)) 'or' (b . z)) '&' (( 'not' (b . z)) 'or' (c . z))) = FALSE ;

              now

                per cases by A16, MARGREL1: 12;

                  case

                   A17: (( 'not' (a . z)) 'or' (b . z)) = FALSE ;

                  (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;

                  hence thesis by A13, A12, A17;

                end;

                  case

                   A18: (( 'not' (b . z)) 'or' (c . z)) = FALSE ;

                  (c . z) = TRUE or (c . z) = FALSE by XBOOLEAN:def 3;

                  hence thesis by A13, A12, A18;

                end;

              end;

              hence thesis;

            end;

              case

               A19: (( 'not' (c . z)) 'or' (a . z)) = FALSE ;

              (a . z) = TRUE or (a . z) = FALSE by XBOOLEAN:def 3;

              hence thesis by A13, A12, A19;

            end;

          end;

          hence thesis;

        end;

        hence thesis;

      end;

      then (((a '&' b) '&' c) 'or' ((( 'not' a) '&' ( 'not' b)) '&' ( 'not' c))) '<' (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a));

      hence thesis by A11, BVFUNC_1: 15;

    end;

    theorem :: BVFUNC_6:156

    for a,b,c be Function of Y, BOOLEAN holds ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c)) = ((a '&' b) '&' c)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c)) = ((((( 'not' a) 'or' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c)) by BVFUNC_4: 8

      .= ((((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c)) by BVFUNC_4: 8

      .= ((((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (( 'not' c) 'or' a)) '&' ((a 'or' b) 'or' c)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' ((( 'not' c) 'or' a) '&' ((a 'or' b) 'or' c))) by BVFUNC_1: 4

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (((( 'not' c) 'or' a) '&' (a 'or' b)) 'or' ((( 'not' c) 'or' a) '&' c))) by BVFUNC_1: 12

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (((( 'not' c) 'or' a) '&' (a 'or' b)) 'or' ((( 'not' c) '&' c) 'or' (a '&' c)))) by BVFUNC_1: 12

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (((( 'not' c) 'or' a) '&' (a 'or' b)) 'or' (( O_el Y) 'or' (a '&' c)))) by BVFUNC_4: 5

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (((( 'not' c) 'or' a) '&' (a 'or' b)) 'or' (a '&' c))) by BVFUNC_1: 9

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' ((a 'or' (( 'not' c) '&' b)) 'or' (a '&' c))) by BVFUNC_1: 11

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' ((a 'or' (a '&' c)) 'or' (( 'not' c) '&' b))) by BVFUNC_1: 8

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (((a '&' ( I_el Y)) 'or' (a '&' c)) 'or' (( 'not' c) '&' b))) by BVFUNC_1: 6

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' ((a '&' (( I_el Y) 'or' c)) 'or' (( 'not' c) '&' b))) by BVFUNC_1: 12

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' ((a '&' ( I_el Y)) 'or' (( 'not' c) '&' b))) by BVFUNC_1: 10

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' (a 'or' (( 'not' c) '&' b))) by BVFUNC_1: 6

      .= (((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c)) '&' ((a 'or' ( 'not' c)) '&' (a 'or' b))) by BVFUNC_1: 11

      .= (((a 'or' b) '&' ((( 'not' a) 'or' b) '&' (( 'not' b) 'or' c))) '&' (a 'or' ( 'not' c))) by BVFUNC_1: 4

      .= ((((a 'or' b) '&' (( 'not' a) 'or' b)) '&' (( 'not' b) 'or' c)) '&' (a 'or' ( 'not' c))) by BVFUNC_1: 4

      .= ((((a '&' ( 'not' a)) 'or' b) '&' (( 'not' b) 'or' c)) '&' (a 'or' ( 'not' c))) by BVFUNC_1: 11

      .= (((( O_el Y) 'or' b) '&' (( 'not' b) 'or' c)) '&' (a 'or' ( 'not' c))) by BVFUNC_4: 5

      .= ((b '&' (( 'not' b) 'or' c)) '&' (a 'or' ( 'not' c))) by BVFUNC_1: 9

      .= (((b '&' ( 'not' b)) 'or' (b '&' c)) '&' (a 'or' ( 'not' c))) by BVFUNC_1: 12

      .= ((( O_el Y) 'or' (b '&' c)) '&' (a 'or' ( 'not' c))) by BVFUNC_4: 5

      .= ((b '&' c) '&' (a 'or' ( 'not' c))) by BVFUNC_1: 9

      .= (((b '&' c) '&' a) 'or' ((b '&' c) '&' ( 'not' c))) by BVFUNC_1: 12

      .= (((b '&' c) '&' a) 'or' (b '&' (c '&' ( 'not' c)))) by BVFUNC_1: 4

      .= (((b '&' c) '&' a) 'or' (b '&' ( O_el Y))) by BVFUNC_4: 5

      .= (((b '&' c) '&' a) 'or' ( O_el Y)) by BVFUNC_1: 5

      .= ((b '&' c) '&' a) by BVFUNC_1: 9

      .= ((a '&' b) '&' c) by BVFUNC_1: 4;

      hence thesis;

    end;



     Lm4: for a,b,c be Function of Y, BOOLEAN holds ((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) '<' ((a 'or' b) '&' ( 'not' ((a '&' b) '&' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) . z) = (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) . z) 'or' (((a '&' b) '&' ( 'not' c)) . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a '&' ( 'not' b)) '&' c) . z)) 'or' (((a '&' b) '&' ( 'not' c)) . z)) by BVFUNC_1:def 4

      .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a '&' ( 'not' b)) '&' c) . z)) 'or' (((a '&' b) . z) '&' (( 'not' c) . z))) by MARGREL1:def 20

      .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a '&' ( 'not' b)) '&' c) . z)) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

      .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a '&' ( 'not' b)) . z) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

      .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a . z) '&' (( 'not' b) . z)) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

      .= (((((( 'not' a) '&' b) . z) '&' (c . z)) 'or' (((a . z) '&' (( 'not' b) . z)) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

      .= (((((( 'not' a) . z) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' (( 'not' b) . z)) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

      .= ((((( 'not' (a . z)) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' (( 'not' b) . z)) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 19

      .= ((((( 'not' (a . z)) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' ( 'not' (b . z))) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 19

      .= ((((( 'not' (a . z)) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' ( 'not' (b . z))) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' ( 'not' (c . z)))) by MARGREL1:def 19;

      assume

       A2: (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) . z) = TRUE ;

      now



         A3: (((a 'or' b) '&' ( 'not' ((a '&' b) '&' c))) . z) = (((a 'or' b) '&' ((( 'not' a) 'or' ( 'not' b)) 'or' ( 'not' c))) . z) by BVFUNC_5: 37

        .= (((a 'or' b) . z) '&' (((( 'not' a) 'or' ( 'not' b)) 'or' ( 'not' c)) . z)) by MARGREL1:def 20

        .= (((a . z) 'or' (b . z)) '&' (((( 'not' a) 'or' ( 'not' b)) 'or' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= (((a . z) 'or' (b . z)) '&' (((( 'not' a) 'or' ( 'not' b)) . z) 'or' (( 'not' c) . z))) by BVFUNC_1:def 4

        .= (((a . z) 'or' (b . z)) '&' (((( 'not' a) . z) 'or' (( 'not' b) . z)) 'or' (( 'not' c) . z))) by BVFUNC_1:def 4

        .= (((a . z) 'or' (b . z)) '&' ((( 'not' (a . z)) 'or' (( 'not' b) . z)) 'or' (( 'not' c) . z))) by MARGREL1:def 19

        .= (((a . z) 'or' (b . z)) '&' ((( 'not' (a . z)) 'or' ( 'not' (b . z))) 'or' (( 'not' c) . z))) by MARGREL1:def 19

        .= (((a . z) 'or' (b . z)) '&' ((( 'not' (a . z)) 'or' ( 'not' (b . z))) 'or' ( 'not' (c . z)))) by MARGREL1:def 19;

        assume (((a 'or' b) '&' ( 'not' ((a '&' b) '&' c))) . z) <> TRUE ;

        then

         A4: (((a . z) 'or' (b . z)) '&' ((( 'not' (a . z)) 'or' ( 'not' (b . z))) 'or' ( 'not' (c . z)))) = FALSE by A3, XBOOLEAN:def 3;

        now

          per cases by A4, MARGREL1: 12;

            case

             A5: ((a . z) 'or' (b . z)) = FALSE ;

            (b . z) = TRUE or (b . z) = FALSE by XBOOLEAN:def 3;

            hence thesis by A2, A1, A5;

          end;

            case

             A6: ((( 'not' (a . z)) 'or' ( 'not' (b . z))) 'or' ( 'not' (c . z))) = FALSE ;



             A7: ( 'not' (b . z)) = TRUE or ( 'not' (b . z)) = FALSE by XBOOLEAN:def 3;

            (( 'not' (a . z)) 'or' ( 'not' (b . z))) = TRUE or (( 'not' (a . z)) 'or' ( 'not' (b . z))) = FALSE by XBOOLEAN:def 3;

            hence thesis by A2, A1, A6, A7;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:157

    for a,b,c be Function of Y, BOOLEAN holds ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c))) = ((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;



       A1: ((((a 'or' b) '&' (b 'or' c)) '&' ( 'not' ((a '&' b) '&' c))) '&' ((c 'or' a) '&' ( 'not' ((a '&' b) '&' c)))) = (((((a 'or' b) '&' (b 'or' c)) '&' ( 'not' ((a '&' b) '&' c))) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4

      .= (((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c))) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4

      .= ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' (( 'not' ((a '&' b) '&' c)) '&' ( 'not' ((a '&' b) '&' c)))) by BVFUNC_1: 4

      .= ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c)));

      for z be Element of Y st (((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c))) . z) = TRUE holds (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) . z) = TRUE

      proof

        let z be Element of Y;



         A2: (((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c))) . z) = (((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) . z) '&' (( 'not' ((a '&' b) '&' c)) . z)) by MARGREL1:def 20

        .= (((((a 'or' b) '&' (b 'or' c)) . z) '&' ((c 'or' a) . z)) '&' (( 'not' ((a '&' b) '&' c)) . z)) by MARGREL1:def 20

        .= (((((a 'or' b) . z) '&' ((b 'or' c) . z)) '&' ((c 'or' a) . z)) '&' (( 'not' ((a '&' b) '&' c)) . z)) by MARGREL1:def 20

        .= (((((a 'or' b) . z) '&' ((b 'or' c) . z)) '&' ((c 'or' a) . z)) '&' (((( 'not' a) 'or' ( 'not' b)) 'or' ( 'not' c)) . z)) by BVFUNC_5: 37

        .= (((((a . z) 'or' (b . z)) '&' ((b 'or' c) . z)) '&' ((c 'or' a) . z)) '&' (((( 'not' a) 'or' ( 'not' b)) 'or' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= (((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c 'or' a) . z)) '&' (((( 'not' a) 'or' ( 'not' b)) 'or' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= (((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c . z) 'or' (a . z))) '&' (((( 'not' a) 'or' ( 'not' b)) 'or' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= (((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c . z) 'or' (a . z))) '&' (((( 'not' a) 'or' ( 'not' b)) . z) 'or' (( 'not' c) . z))) by BVFUNC_1:def 4

        .= (((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c . z) 'or' (a . z))) '&' (((( 'not' a) . z) 'or' (( 'not' b) . z)) 'or' (( 'not' c) . z))) by BVFUNC_1:def 4

        .= (((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c . z) 'or' (a . z))) '&' ((( 'not' (a . z)) 'or' (( 'not' b) . z)) 'or' (( 'not' c) . z))) by MARGREL1:def 19

        .= (((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c . z) 'or' (a . z))) '&' ((( 'not' (a . z)) 'or' ( 'not' (b . z))) 'or' (( 'not' c) . z))) by MARGREL1:def 19

        .= (((((a . z) 'or' (b . z)) '&' ((b . z) 'or' (c . z))) '&' ((c . z) 'or' (a . z))) '&' ((( 'not' (a . z)) 'or' ( 'not' (b . z))) 'or' ( 'not' (c . z)))) by MARGREL1:def 19;

        assume

         A3: (((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c))) . z) = TRUE ;

        now



           A4: (((a . z) '&' (b . z)) '&' ( 'not' (c . z))) = TRUE or (((a . z) '&' (b . z)) '&' ( 'not' (c . z))) = FALSE by XBOOLEAN:def 3;

          assume

           A5: (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) . z) <> TRUE ;



           A6: (((a . z) '&' ( 'not' (b . z))) '&' (c . z)) = TRUE or (((a . z) '&' ( 'not' (b . z))) '&' (c . z)) = FALSE by XBOOLEAN:def 3;



           A7: (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) . z) = (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) . z) 'or' (((a '&' b) '&' ( 'not' c)) . z)) by BVFUNC_1:def 4

          .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a '&' ( 'not' b)) '&' c) . z)) 'or' (((a '&' b) '&' ( 'not' c)) . z)) by BVFUNC_1:def 4

          .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a '&' ( 'not' b)) '&' c) . z)) 'or' (((a '&' b) . z) '&' (( 'not' c) . z))) by MARGREL1:def 20

          .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a '&' ( 'not' b)) '&' c) . z)) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

          .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a '&' ( 'not' b)) . z) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

          .= (((((( 'not' a) '&' b) '&' c) . z) 'or' (((a . z) '&' (( 'not' b) . z)) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

          .= (((((( 'not' a) '&' b) . z) '&' (c . z)) 'or' (((a . z) '&' (( 'not' b) . z)) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

          .= (((((( 'not' a) . z) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' (( 'not' b) . z)) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 20

          .= ((((( 'not' (a . z)) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' (( 'not' b) . z)) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 19

          .= ((((( 'not' (a . z)) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' ( 'not' (b . z))) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' (( 'not' c) . z))) by MARGREL1:def 19

          .= ((((( 'not' (a . z)) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' ( 'not' (b . z))) '&' (c . z))) 'or' (((a . z) '&' (b . z)) '&' ( 'not' (c . z)))) by MARGREL1:def 19;



           A8: (((( 'not' (a . z)) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' ( 'not' (b . z))) '&' (c . z))) = TRUE or (((( 'not' (a . z)) '&' (b . z)) '&' (c . z)) 'or' (((a . z) '&' ( 'not' (b . z))) '&' (c . z))) = FALSE by XBOOLEAN:def 3;

          now

            per cases by A5, A7, A8, A6, MARGREL1: 12;

              case

               A9: (( 'not' (a . z)) '&' (b . z)) = FALSE ;

              now

                per cases by A9, MARGREL1: 12;

                  case

                   A10: ( 'not' (a . z)) = FALSE ;

                  now

                    per cases by A5, A7, A6, A10, MARGREL1: 12;

                      case ( 'not' (b . z)) = FALSE ;

                      hence thesis by A3, A2, A7, A10;

                    end;

                      case (c . z) = FALSE ;

                      hence thesis by A3, A2, A7, A10;

                    end;

                  end;

                  hence thesis;

                end;

                  case

                   A11: (b . z) = FALSE ;

                  now

                    per cases by A5, A7, A6, A11, MARGREL1: 12;

                      case (a . z) = FALSE ;

                      hence thesis by A3, A2, A11;

                    end;

                      case (c . z) = FALSE ;

                      hence thesis by A3, A2, A11;

                    end;

                  end;

                  hence thesis;

                end;

              end;

              hence thesis;

            end;

              case

               A12: (c . z) = FALSE ;

              now

                per cases by A5, A7, A4, A12, MARGREL1: 12;

                  case (a . z) = FALSE ;

                  hence thesis by A3, A2, A12;

                end;

                  case (b . z) = FALSE ;

                  hence thesis by A3, A2, A12;

                end;

              end;

              hence thesis;

            end;

          end;

          hence thesis;

        end;

        hence thesis;

      end;

      then

       A13: ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c))) '<' ((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c)));

      ((((( 'not' c) '&' a) '&' b) 'or' ((c '&' ( 'not' a)) '&' b)) 'or' ((c '&' a) '&' ( 'not' b))) '<' ((c 'or' a) '&' ( 'not' ((c '&' a) '&' b))) by Lm4;

      then ((((( 'not' c) '&' a) '&' b) 'or' ((c '&' ( 'not' a)) '&' b)) 'or' ((c '&' a) '&' ( 'not' b))) '<' ((c 'or' a) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4;

      then ((((a '&' b) '&' ( 'not' c)) 'or' ((c '&' ( 'not' a)) '&' b)) 'or' ((c '&' a) '&' ( 'not' b))) '<' ((c 'or' a) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4;

      then ((((a '&' b) '&' ( 'not' c)) 'or' ((( 'not' a) '&' b) '&' c)) 'or' ((c '&' a) '&' ( 'not' b))) '<' ((c 'or' a) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4;

      then ((((a '&' b) '&' ( 'not' c)) 'or' ((( 'not' a) '&' b) '&' c)) 'or' ((a '&' ( 'not' b)) '&' c)) '<' ((c 'or' a) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4;

      then ((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) '<' ((c 'or' a) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 8;

      then

       A14: (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) 'imp' ((c 'or' a) '&' ( 'not' ((a '&' b) '&' c)))) = ( I_el Y) by BVFUNC_1: 16;

      ((((( 'not' b) '&' c) '&' a) 'or' ((b '&' ( 'not' c)) '&' a)) 'or' ((b '&' c) '&' ( 'not' a))) '<' ((b 'or' c) '&' ( 'not' ((b '&' c) '&' a))) by Lm4;

      then ((((( 'not' b) '&' c) '&' a) 'or' ((b '&' ( 'not' c)) '&' a)) 'or' ((b '&' c) '&' ( 'not' a))) '<' ((b 'or' c) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4;

      then ((((( 'not' b) '&' c) '&' a) 'or' ((b '&' ( 'not' c)) '&' a)) 'or' ((( 'not' a) '&' b) '&' c)) '<' ((b 'or' c) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4;

      then ((((a '&' ( 'not' b)) '&' c) 'or' ((b '&' ( 'not' c)) '&' a)) 'or' ((( 'not' a) '&' b) '&' c)) '<' ((b 'or' c) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4;

      then ((((a '&' ( 'not' b)) '&' c) 'or' ((a '&' b) '&' ( 'not' c))) 'or' ((( 'not' a) '&' b) '&' c)) '<' ((b 'or' c) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 4;

      then ((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) '<' ((b 'or' c) '&' ( 'not' ((a '&' b) '&' c))) by BVFUNC_1: 8;

      then

       A15: (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) 'imp' ((b 'or' c) '&' ( 'not' ((a '&' b) '&' c)))) = ( I_el Y) by BVFUNC_1: 16;

      ((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) '<' ((a 'or' b) '&' ( 'not' ((a '&' b) '&' c))) by Lm4;

      then (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) 'imp' ((a 'or' b) '&' ( 'not' ((a '&' b) '&' c)))) = ( I_el Y) by BVFUNC_1: 16;

      then (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) 'imp' (((a 'or' b) '&' ( 'not' ((a '&' b) '&' c))) '&' ((b 'or' c) '&' ( 'not' ((a '&' b) '&' c))))) = ( I_el Y) by A15, th18;

      then (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) 'imp' ((((a 'or' b) '&' ( 'not' ((a '&' b) '&' c))) '&' (b 'or' c)) '&' ( 'not' ((a '&' b) '&' c)))) = ( I_el Y) by BVFUNC_1: 4;

      then (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) 'imp' ((((a 'or' b) '&' (b 'or' c)) '&' ( 'not' ((a '&' b) '&' c))) '&' ( 'not' ((a '&' b) '&' c)))) = ( I_el Y) by BVFUNC_1: 4;

      then (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) 'imp' (((a 'or' b) '&' (b 'or' c)) '&' (( 'not' ((a '&' b) '&' c)) '&' ( 'not' ((a '&' b) '&' c))))) = ( I_el Y) by BVFUNC_1: 4;

      then (((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) 'imp' ((((a 'or' b) '&' (b 'or' c)) '&' ( 'not' ((a '&' b) '&' c))) '&' ((c 'or' a) '&' ( 'not' ((a '&' b) '&' c))))) = ( I_el Y) by A14, th18;

      then ((((( 'not' a) '&' b) '&' c) 'or' ((a '&' ( 'not' b)) '&' c)) 'or' ((a '&' b) '&' ( 'not' c))) '<' ((((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ( 'not' ((a '&' b) '&' c))) by A1, BVFUNC_1: 16;

      hence thesis by A13, BVFUNC_1: 15;

    end;

    theorem :: BVFUNC_6:158

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' (a 'imp' (b '&' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (b 'imp' c)) . z) = (((a 'imp' b) . z) '&' ((b 'imp' c) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((b 'imp' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (b 'imp' c)) . z) = TRUE ;

      now



         A3: ((a 'imp' (b '&' c)) . z) = ((( 'not' a) 'or' (b '&' c)) . z) by BVFUNC_4: 8

        .= ((( 'not' a) . z) 'or' ((b '&' c) . z)) by BVFUNC_1:def 4

        .= ((( 'not' a) . z) 'or' ((b . z) '&' (c . z))) by MARGREL1:def 20

        .= (( 'not' (a . z)) 'or' ((b . z) '&' (c . z))) by MARGREL1:def 19;

        assume

         A4: ((a 'imp' (b '&' c)) . z) <> TRUE ;

        ( 'not' (a . z)) = TRUE or ( 'not' (a . z)) = FALSE by XBOOLEAN:def 3;

        then

         A5: (( 'not' a) . z) = FALSE by A4, A3, MARGREL1:def 19;



         A6: ((b . z) '&' (c . z)) = TRUE or ((b . z) '&' (c . z)) = FALSE by XBOOLEAN:def 3;

        now

          per cases by A4, A3, A6, MARGREL1: 12;

            case (b . z) = FALSE ;

            hence thesis by A2, A1, A5;

          end;

            case (c . z) = FALSE ;



            then (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) = ((b . z) '&' ( 'not' (b . z))) by A5, MARGREL1:def 19

            .= FALSE by XBOOLEAN: 138;

            hence thesis by A2, A1;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:159

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' ((a 'or' b) 'imp' c)

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (b 'imp' c)) . z) = (((a 'imp' b) . z) '&' ((b 'imp' c) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((b 'imp' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (b 'imp' c)) . z) = TRUE ;

      now



         A3: (c . z) = TRUE or (c . z) = FALSE by XBOOLEAN:def 3;

        assume

         A4: (((a 'or' b) 'imp' c) . z) <> TRUE ;



         A5: (((a 'or' b) 'imp' c) . z) = ((( 'not' (a 'or' b)) 'or' c) . z) by BVFUNC_4: 8

        .= (((( 'not' a) '&' ( 'not' b)) 'or' c) . z) by BVFUNC_1: 13

        .= (((( 'not' a) '&' ( 'not' b)) . z) 'or' (c . z)) by BVFUNC_1:def 4

        .= (((( 'not' a) . z) '&' (( 'not' b) . z)) 'or' (c . z)) by MARGREL1:def 20;



         A6: ((( 'not' a) . z) '&' (( 'not' b) . z)) = TRUE or ((( 'not' a) . z) '&' (( 'not' b) . z)) = FALSE by XBOOLEAN:def 3;

        now

          per cases by A4, A5, A6, MARGREL1: 12;

            case (( 'not' a) . z) = FALSE ;



            then (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) = ((b . z) '&' ( 'not' (b . z))) by A4, A5, A3, MARGREL1:def 19

            .= FALSE by XBOOLEAN: 138;

            hence thesis by A2, A1;

          end;

            case (( 'not' b) . z) = FALSE ;



            then (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) = (((( 'not' a) . z) 'or' (b . z)) '&' FALSE ) by A4, A5, XBOOLEAN:def 3

            .= FALSE ;

            hence thesis by A2, A1;

          end;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:160



     Th19: for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' (a 'imp' (b 'or' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (b 'imp' c)) . z) = (((a 'imp' b) . z) '&' ((b 'imp' c) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((b 'imp' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (b 'imp' c)) . z) = TRUE ;

      now

        assume ((a 'imp' (b 'or' c)) . z) <> TRUE ;



         A3: (( 'not' a) . z) = TRUE or (( 'not' a) . z) = FALSE by XBOOLEAN:def 3;



         A4: ((b . z) 'or' (c . z)) = TRUE or ((b . z) 'or' (c . z)) = FALSE by XBOOLEAN:def 3;



         A5: (c . z) = TRUE or (c . z) = FALSE by XBOOLEAN:def 3;

        ((a 'imp' (b 'or' c)) . z) = ((( 'not' a) 'or' (b 'or' c)) . z) by BVFUNC_4: 8

        .= ((( 'not' a) . z) 'or' ((b 'or' c) . z)) by BVFUNC_1:def 4

        .= ((( 'not' a) . z) 'or' ((b . z) 'or' (c . z))) by BVFUNC_1:def 4;

        hence thesis by A2, A1, A3, A4, A5;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:161



     Th20: for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' (a 'imp' (b 'or' ( 'not' c)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (b 'imp' c)) . z) = (((a 'imp' b) . z) '&' ((b 'imp' c) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((b 'imp' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (b 'imp' c)) . z) = TRUE ;

      now

        assume ((a 'imp' (b 'or' ( 'not' c))) . z) <> TRUE ;



         A3: (( 'not' a) . z) = TRUE or (( 'not' a) . z) = FALSE by XBOOLEAN:def 3;



         A4: ((b . z) 'or' (( 'not' c) . z)) = TRUE or ((b . z) 'or' (( 'not' c) . z)) = FALSE by XBOOLEAN:def 3;



         A5: (( 'not' c) . z) = TRUE or (( 'not' c) . z) = FALSE by XBOOLEAN:def 3;

        ((a 'imp' (b 'or' ( 'not' c))) . z) = ((( 'not' a) 'or' (b 'or' ( 'not' c))) . z) by BVFUNC_4: 8

        .= ((( 'not' a) . z) 'or' ((b 'or' ( 'not' c)) . z)) by BVFUNC_1:def 4

        .= ((( 'not' a) . z) 'or' ((b . z) 'or' (( 'not' c) . z))) by BVFUNC_1:def 4;

        hence thesis by A2, A1, A3, A4, A5;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:162



     Th21: for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' (b 'imp' (c 'or' a))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (b 'imp' c)) . z) = (((a 'imp' b) . z) '&' ((b 'imp' c) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((b 'imp' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (b 'imp' c)) . z) = TRUE ;

      now

        assume ((b 'imp' (c 'or' a)) . z) <> TRUE ;



         A3: (( 'not' b) . z) = TRUE or (( 'not' b) . z) = FALSE by XBOOLEAN:def 3;



         A4: ((c . z) 'or' (a . z)) = TRUE or ((c . z) 'or' (a . z)) = FALSE by XBOOLEAN:def 3;



         A5: (a . z) = TRUE or (a . z) = FALSE by XBOOLEAN:def 3;

        ((b 'imp' (c 'or' a)) . z) = ((( 'not' b) 'or' (c 'or' a)) . z) by BVFUNC_4: 8

        .= ((( 'not' b) . z) 'or' ((c 'or' a) . z)) by BVFUNC_1:def 4

        .= ((( 'not' b) . z) 'or' ((c . z) 'or' (a . z))) by BVFUNC_1:def 4;

        hence thesis by A2, A1, A3, A4, A5;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:163



     Th22: for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' (b 'imp' (c 'or' ( 'not' a)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      let z be Element of Y;



       A1: (((a 'imp' b) '&' (b 'imp' c)) . z) = (((a 'imp' b) . z) '&' ((b 'imp' c) . z)) by MARGREL1:def 20

      .= (((( 'not' a) 'or' b) . z) '&' ((b 'imp' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) 'or' b) . z) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_4: 8

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) 'or' c) . z)) by BVFUNC_1:def 4

      .= (((( 'not' a) . z) 'or' (b . z)) '&' ((( 'not' b) . z) 'or' (c . z))) by BVFUNC_1:def 4;

      assume

       A2: (((a 'imp' b) '&' (b 'imp' c)) . z) = TRUE ;

      now

        assume ((b 'imp' (c 'or' ( 'not' a))) . z) <> TRUE ;



         A3: (( 'not' b) . z) = TRUE or (( 'not' b) . z) = FALSE by XBOOLEAN:def 3;



         A4: ((c . z) 'or' (( 'not' a) . z)) = TRUE or ((c . z) 'or' (( 'not' a) . z)) = FALSE by XBOOLEAN:def 3;



         A5: (( 'not' a) . z) = TRUE or (( 'not' a) . z) = FALSE by XBOOLEAN:def 3;

        ((b 'imp' (c 'or' ( 'not' a))) . z) = ((( 'not' b) 'or' (c 'or' ( 'not' a))) . z) by BVFUNC_4: 8

        .= ((( 'not' b) . z) 'or' ((c 'or' ( 'not' a)) . z)) by BVFUNC_1:def 4

        .= ((( 'not' b) . z) 'or' ((c . z) 'or' (( 'not' a) . z))) by BVFUNC_1:def 4;

        hence thesis by A2, A1, A3, A4, A5;

      end;

      hence thesis;

    end;

    theorem :: BVFUNC_6:164

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' ((a 'imp' b) '&' (b 'imp' (c 'or' a)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      ((a 'imp' b) '&' (b 'imp' c)) '<' (b 'imp' (c 'or' a)) by Th21;

      then

       A1: (((a 'imp' b) '&' (b 'imp' c)) 'imp' (b 'imp' (c 'or' a))) = ( I_el Y) by BVFUNC_1: 16;

      (((a 'imp' b) '&' (b 'imp' c)) 'imp' (a 'imp' b)) = ( I_el Y) by Th38;

      then (((a 'imp' b) '&' (b 'imp' c)) 'imp' ((a 'imp' b) '&' (b 'imp' (c 'or' a)))) = ( I_el Y) by A1, th18;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:165

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' ((a 'imp' (b 'or' ( 'not' c))) '&' (b 'imp' c))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      ((a 'imp' b) '&' (b 'imp' c)) '<' (a 'imp' (b 'or' ( 'not' c))) by Th20;

      then

       A1: (((a 'imp' b) '&' (b 'imp' c)) 'imp' (a 'imp' (b 'or' ( 'not' c)))) = ( I_el Y) by BVFUNC_1: 16;

      (((a 'imp' b) '&' (b 'imp' c)) 'imp' (b 'imp' c)) = ( I_el Y) by Th38;

      then (((a 'imp' b) '&' (b 'imp' c)) 'imp' ((a 'imp' (b 'or' ( 'not' c))) '&' (b 'imp' c))) = ( I_el Y) by A1, th18;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:166

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' ((a 'imp' (b 'or' c)) '&' (b 'imp' (c 'or' a)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      ((a 'imp' b) '&' (b 'imp' c)) '<' (b 'imp' (c 'or' a)) by Th21;

      then

       A1: (((a 'imp' b) '&' (b 'imp' c)) 'imp' (b 'imp' (c 'or' a))) = ( I_el Y) by BVFUNC_1: 16;

      ((a 'imp' b) '&' (b 'imp' c)) '<' (a 'imp' (b 'or' c)) by Th19;

      then (((a 'imp' b) '&' (b 'imp' c)) 'imp' (a 'imp' (b 'or' c))) = ( I_el Y) by BVFUNC_1: 16;

      then (((a 'imp' b) '&' (b 'imp' c)) 'imp' ((a 'imp' (b 'or' c)) '&' (b 'imp' (c 'or' a)))) = ( I_el Y) by A1, th18;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:167

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' ((a 'imp' (b 'or' ( 'not' c))) '&' (b 'imp' (c 'or' a)))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      ((a 'imp' b) '&' (b 'imp' c)) '<' (b 'imp' (c 'or' a)) by Th21;

      then

       A1: (((a 'imp' b) '&' (b 'imp' c)) 'imp' (b 'imp' (c 'or' a))) = ( I_el Y) by BVFUNC_1: 16;

      ((a 'imp' b) '&' (b 'imp' c)) '<' (a 'imp' (b 'or' ( 'not' c))) by Th20;

      then (((a 'imp' b) '&' (b 'imp' c)) 'imp' (a 'imp' (b 'or' ( 'not' c)))) = ( I_el Y) by BVFUNC_1: 16;

      then (((a 'imp' b) '&' (b 'imp' c)) 'imp' ((a 'imp' (b 'or' ( 'not' c))) '&' (b 'imp' (c 'or' a)))) = ( I_el Y) by A1, th18;

      hence thesis by BVFUNC_1: 16;

    end;

    theorem :: BVFUNC_6:168

    for a,b,c be Function of Y, BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '<' ((a 'imp' (b 'or' ( 'not' c))) '&' (b 'imp' (c 'or' ( 'not' a))))

    proof

      let a,b,c be Function of Y, BOOLEAN ;

      ((a 'imp' b) '&' (b 'imp' c)) '<' (b 'imp' (c 'or' ( 'not' a))) by Th22;

      then

       A1: (((a 'imp' b) '&' (b 'imp' c)) 'imp' (b 'imp' (c 'or' ( 'not' a)))) = ( I_el Y) by BVFUNC_1: 16;

      ((a 'imp' b) '&' (b 'imp' c)) '<' (a 'imp' (b 'or' ( 'not' c))) by Th20;

      then (((a 'imp' b) '&' (b 'imp' c)) 'imp' (a 'imp' (b 'or' ( 'not' c)))) = ( I_el Y) by BVFUNC_1: 16;

      then (((a 'imp' b) '&' (b 'imp' c)) 'imp' ((a 'imp' (b 'or' ( 'not' c))) '&' (b 'imp' (c 'or' ( 'not' a))))) = ( I_el Y) by A1, th18;

      hence thesis by BVFUNC_1: 16;

    end;