let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN st a 'imp' c = I_el Y & b 'imp' c = I_el Y holds
(a 'or' b) 'imp' c = I_el Y
let a, b, c be Function of Y,BOOLEAN; :: thesis: ( a 'imp' c = I_el Y & b 'imp' c = I_el Y implies (a 'or' b) 'imp' c = I_el Y )
assume that
A1: a 'imp' c = I_el Y and
A2: b 'imp' c = I_el Y ; :: thesis: (a 'or' b) 'imp' c = I_el Y
for x being Element of Y holds ((a 'or' b) 'imp' c) . x = TRUE

proof

let x be Element of Y; :: thesis: ((a 'or' b) 'imp' c) . x = TRUE
(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 ((a 'or' b) 'imp' c) . x = TRUE ; :: thesis: verum

end;

hence (a 'or' b) 'imp' c = I_el Y by BVFUNC_1:def 11; :: thesis: verum