let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
let PA be a_partition of Y; :: thesis: (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE or (Ex ((a 'imp' b),PA,G)) . z = TRUE )
A1: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
assume ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE ; :: thesis: (Ex ((a 'imp' b),PA,G)) . z = TRUE
then A2: ('not' ((Ex (a,PA,G)) . z)) 'or' ((Ex (b,PA,G)) . z) = TRUE by BVFUNC_1:def 8;
A3: ( 'not' ((Ex (a,PA,G)) . z) = TRUE or 'not' ((Ex (a,PA,G)) . z) = FALSE ) by XBOOLEAN:def 3;

now :: thesis: ( ( 'not' ((Ex (a,PA,G)) . z) = TRUE & (Ex ((a 'imp' b),PA,G)) . z = TRUE ) or ( (Ex (b,PA,G)) . z = TRUE & (Ex ((a 'imp' b),PA,G)) . z = TRUE ) )

per cases ( 'not' ((Ex (a,PA,G)) . z) = TRUE or (Ex (b,PA,G)) . z = TRUE ) by A2, A3, BINARITH:3;

case 'not' ((Ex (a,PA,G)) . z) = TRUE ; :: thesis: (Ex ((a 'imp' b),PA,G)) . z = TRUE

then A4: a . z <> TRUE by A1, BVFUNC_1:def 17, MARGREL1:11;
(a 'imp' b) . z = ('not' (a . z)) 'or' (b . z) by BVFUNC_1:def 8
.= TRUE 'or' (b . z) by A4, MARGREL1:11, XBOOLEAN:def 3
.= TRUE by BINARITH:10 ;
hence (Ex ((a 'imp' b),PA,G)) . z = TRUE by A1, BVFUNC_1:def 17; :: thesis: verum

end;

case (Ex (b,PA,G)) . z = TRUE ; :: thesis: (Ex ((a 'imp' b),PA,G)) . z = TRUE

then consider x1 being Element of Y such that
A5: x1 in EqClass (z,(CompF (PA,G))) and
A6: b . x1 = TRUE by BVFUNC_1:def 17;
(a 'imp' b) . x1 = ('not' (a . x1)) 'or' (b . x1) by BVFUNC_1:def 8
.= TRUE by A6, BINARITH:10 ;
hence (Ex ((a 'imp' b),PA,G)) . z = TRUE by A5, BVFUNC_1:def 17; :: thesis: verum

end;

end;

end;

hence (Ex ((a 'imp' b),PA,G)) . z = TRUE ; :: thesis: verum