let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) = ((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)
let a, b, c be Element of Funcs Y,BOOLEAN ; :: thesis: (a 'imp' b) '&' (b 'imp' c) = ((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)
A1: for x being Element of Y holds ((a 'imp' b) '&' (b 'imp' c)) . x = (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x

proof

let x be Element of Y; :: thesis: ((a 'imp' b) '&' (b 'imp' c)) . x = (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x
A2: (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x = (((a 'imp' b) '&' (b 'imp' c)) . x) '&' ((a 'imp' c) . x) by MARGREL1:def 21
.= (((a 'imp' b) . x) '&' ((b 'imp' c) . x)) '&' ((a 'imp' c) . x) by MARGREL1:def 21
.= ((('not' (a . x)) 'or' (b . x)) '&' ((b 'imp' c) . x)) '&' ((a 'imp' c) . x) by BVFUNC_1:def 11
.= ((('not' (a . x)) 'or' (b . x)) '&' (('not' (b . x)) 'or' (c . x))) '&' ((a 'imp' c) . x) by BVFUNC_1:def 11
.= ((('not' (a . x)) 'or' (b . x)) '&' (('not' (b . x)) 'or' (c . x))) '&' (('not' (a . x)) 'or' (c . x)) by BVFUNC_1:def 11
.= (((('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 ;
A3: ((a 'imp' b) '&' (b 'imp' c)) . x = ((a 'imp' b) . x) '&' ((b 'imp' c) . x) by MARGREL1:def 21
.= (('not' (a . x)) 'or' (b . x)) '&' ((b 'imp' c) . x) by BVFUNC_1:def 11
.= (('not' (a . x)) 'or' (b . x)) '&' (('not' (b . x)) 'or' (c . x)) by BVFUNC_1:def 11 ;
A4: (('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 11
.= ((a 'imp' b) . x) '&' ((b 'imp' c) . x) by BVFUNC_1:def 11
.= ((a 'imp' b) '&' (b 'imp' c)) . x by MARGREL1:def 21 ;

now

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: ((a 'imp' b) '&' (b 'imp' c)) . x = (((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 A2, MARGREL1:41
.= FALSE 'or' (((('not' (a . x)) 'or' (b . x)) '&' (('not' (b . x)) 'or' (c . x))) '&' TRUE ) by MARGREL1:49
.= FALSE 'or' ((('not' (a . x)) 'or' (b . x)) '&' (('not' (b . x)) 'or' (c . x))) by MARGREL1:50
.= ((a 'imp' b) '&' (b 'imp' c)) . x by A4, BINARITH:7 ;
hence ((a 'imp' b) '&' (b 'imp' c)) . x = (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x ; :: thesis: verum

end;

case A5: ( a . x = TRUE & c . x = FALSE ) ; :: thesis: ((a 'imp' b) '&' (b 'imp' c)) . x = (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x

then A6: (((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 A2, MARGREL1:41
.= FALSE by MARGREL1:49 ;
((a 'imp' b) '&' (b 'imp' c)) . x = (FALSE 'or' (b . x)) '&' (('not' (b . x)) 'or' FALSE ) by A3, A5, MARGREL1:41
.= (FALSE 'or' (b . x)) '&' ('not' (b . x)) by BINARITH:7
.= (b . x) '&' ('not' (b . x)) by BINARITH:7
.= FALSE by XBOOLEAN:138 ;
hence ((a 'imp' b) '&' (b 'imp' c)) . x = (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x by A6; :: thesis: verum

end;

case ( a . x = FALSE & c . x = TRUE ) ; :: thesis: ((a 'imp' b) '&' (b 'imp' c)) . x = (((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 A2, MARGREL1:41
.= ((a 'imp' b) '&' (b 'imp' c)) . x by A4, MARGREL1:50 ;
hence ((a 'imp' b) '&' (b 'imp' c)) . x = (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x ; :: thesis: verum

end;

case ( a . x = FALSE & c . x = FALSE ) ; :: thesis: ((a 'imp' b) '&' (b 'imp' c)) . x = (((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 A2, MARGREL1:41
.= ((('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 MARGREL1:50
.= ((('not' (a . x)) 'or' (b . x)) '&' (('not' (b . x)) 'or' (c . x))) 'or' FALSE by MARGREL1:49
.= ((a 'imp' b) '&' (b 'imp' c)) . x by A4, BINARITH:7 ;
hence ((a 'imp' b) '&' (b 'imp' c)) . x = (((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)) . x ; :: thesis: verum

end;

end;

end;

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

end;

consider k3 being Function such that
A7: ( (a 'imp' b) '&' (b 'imp' c) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A8: ( ((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c) = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
( Y = dom k3 & Y = dom k4 & ( for u being set st u in Y holds
k3 . u = k4 . u ) ) by A1, A7, A8;
hence (a 'imp' b) '&' (b 'imp' c) = ((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c) by A7, A8, FUNCT_1:9; :: thesis: verum