let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN holds a 'imp' a = I_el Y
let a be Function of Y,BOOLEAN; :: thesis: a 'imp' a = I_el Y
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (a 'imp' a) . x = (I_el Y) . x
A1: ( (a 'imp' a) . x = ('not' (a . x)) 'or' (a . x) & (I_el Y) . x = TRUE ) by BVFUNC_1:def 8, BVFUNC_1:def 11;
A2: 'not' FALSE = TRUE by MARGREL1:11;

now :: thesis: ( ( a . x = TRUE & (a 'imp' a) . x = (I_el Y) . x ) or ( a . x = FALSE & (a 'imp' a) . x = (I_el Y) . x ) )

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

case a . x = TRUE ; :: thesis: (a 'imp' a) . x = (I_el Y) . x

hence (a 'imp' a) . x = (I_el Y) . x by A1, BINARITH:10; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: (a 'imp' a) . x = (I_el Y) . x

hence (a 'imp' a) . x = (I_el Y) . x by A2, A1, BINARITH:10; :: thesis: verum

end;

end;

end;

hence (a 'imp' a) . x = (I_el Y) . x ; :: thesis: verum