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 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 Element of Funcs 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 Element of Funcs 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 15 :: thesis: ( not (All (a 'or' b),PA,G) . z = TRUE or ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE )
assume A1: (All (a 'or' b),PA,G) . z = TRUE ; :: thesis: ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE
A2: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;
then (a 'or' b) . z = TRUE by A1, Lm1;
then A3: (a . z) 'or' (b . z) = TRUE by BVFUNC_1:def 7;
A4: ( a . z = TRUE or a . z = FALSE ) by XBOOLEAN:def 3;

per cases ( a . z = TRUE or b . z = TRUE ) by A3, A4, BINARITH:7;

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 A2, BVFUNC_1:def 20;
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 7
.= TRUE by BINARITH:19 ;
:: 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 A2, BVFUNC_1:def 20;
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 7
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

end;