let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) = ((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a 'imp' b) '&' (b 'imp' c) = ((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x)
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 :: thesis: ( ( a . x = TRUE & c . x = TRUE & K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) ) or ( a . x = TRUE & c . x = FALSE & K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) ) or ( a . x = FALSE & c . x = TRUE & K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) ) or ( a . x = FALSE & c . x = FALSE & K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) ) )

per cases ( ( a . x = TRUE & c . x = TRUE ) or ( a . x = TRUE & c . x = FALSE ) or ( a . x = FALSE & c . x = TRUE ) or ( a . x = FALSE & c . x = FALSE ) ) by XBOOLEAN:def 3;

case ( a . x = TRUE & c . x = TRUE ) ; :: thesis: K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x)

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 K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) ; :: thesis: verum

end;

case A4: ( a . x = TRUE & c . x = FALSE ) ; :: thesis: K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x)

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 K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) by A5; :: thesis: verum

end;

case ( a . x = FALSE & c . x = TRUE ) ; :: thesis: K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x)

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 K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) ; :: thesis: verum

end;

case ( a . x = FALSE & c . x = FALSE ) ; :: thesis: K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x)

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 K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) ; :: thesis: verum

end;

end;

end;

hence K11(((a 'imp' b) '&' (b 'imp' c)),x) = K11((((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)),x) ; :: thesis: verum