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 All ((a 'or' b),PA,G) '<' (Ex (a,PA,G)) 'or' (Ex (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 All ((a 'or' b),PA,G) '<' (Ex (a,PA,G)) 'or' (Ex (b,PA,G))
let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All ((a 'or' b),PA,G) '<' (Ex (a,PA,G)) 'or' (Ex (b,PA,G))
let PA be a_partition of Y; :: thesis: All ((a 'or' b),PA,G) '<' (Ex (a,PA,G)) 'or' (Ex (b,PA,G))
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a 'or' b),PA,G)) . z = TRUE or ((Ex (a,PA,G)) 'or' (Ex (b,PA,G))) . z = TRUE )
A1: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
assume (All ((a 'or' b),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) 'or' (Ex (b,PA,G))) . z = TRUE
then (a 'or' b) . z = TRUE by A1, Lm1;
then A2: (a . z) 'or' (b . z) = TRUE by BVFUNC_1:def 4;
A3: ( a . z = TRUE or a . z = FALSE ) by XBOOLEAN:def 3;

per cases ( a . z = TRUE or b . z = TRUE ) by A2, A3, BINARITH:3;

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

then (B_SUP (a,(CompF (PA,G)))) . z = TRUE by A1, BVFUNC_1:def 17;
then (Ex (a,PA,G)) . z = TRUE by BVFUNC_2:def 10;
hence ((Ex (a,PA,G)) 'or' (Ex (b,PA,G))) . z = TRUE 'or' ((Ex (b,PA,G)) . z) by BVFUNC_1:def 4
.= TRUE by BINARITH:10 ;
:: thesis: verum

end;

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

then (B_SUP (b,(CompF (PA,G)))) . z = TRUE by A1, BVFUNC_1:def 17;
then (Ex (b,PA,G)) . z = TRUE by BVFUNC_2:def 10;
hence ((Ex (a,PA,G)) 'or' (Ex (b,PA,G))) . z = ((Ex (a,PA,G)) . z) 'or' TRUE by BVFUNC_1:def 4
.= TRUE by BINARITH:10 ;
:: thesis: verum

end;

end;