let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds a 'or' b '<' (Ex a,PA,G) 'or' (Ex b,PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds a 'or' b '<' (Ex a,PA,G) 'or' (Ex b,PA,G)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds a 'or' b '<' (Ex a,PA,G) 'or' (Ex b,PA,G)
let PA be a_partition of Y; :: thesis: a 'or' b '<' (Ex a,PA,G) 'or' (Ex b,PA,G)
A1: Ex a,PA,G = B_SUP a,(CompF PA,G) by BVFUNC_2:def 10;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (a 'or' b) . z = TRUE or ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE )
A2: Ex b,PA,G = B_SUP b,(CompF PA,G) by BVFUNC_2:def 10;
A3: z in EqClass z,(CompF PA,G) by EQREL_1:def 8;
assume (a 'or' b) . z = TRUE ; :: thesis: ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE
then A4: (a . z) 'or' (b . z) = TRUE by BVFUNC_1:def 7;
A5: ( b . z = TRUE or b . z = FALSE ) by XBOOLEAN:def 3;

per cases ( a . z = TRUE or b . z = TRUE ) by A4, A5, BINARITH:7;

suppose A6: a . z = TRUE ; :: thesis: ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE

thus ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = ((Ex a,PA,G) . z) 'or' ((Ex b,PA,G) . z) by BVFUNC_1:def 7
.= TRUE 'or' ((Ex b,PA,G) . z) by A1, A3, A6, BVFUNC_1:def 20
.= TRUE by BINARITH:19 ; :: thesis: verum

end;

suppose A7: b . z = TRUE ; :: thesis: ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE

thus ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = ((Ex a,PA,G) . z) 'or' ((Ex b,PA,G) . z) by BVFUNC_1:def 7
.= ((Ex a,PA,G) . z) 'or' TRUE by A2, A3, A7, BVFUNC_1:def 20
.= TRUE by BINARITH:19 ; :: thesis: verum

end;

end;