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

now :: thesis: ( ( (All (a,PA,G)) . z = TRUE & (All ((a 'or' b),PA,G)) . z = TRUE ) or ( (All (b,PA,G)) . z = TRUE & (All ((a 'or' b),PA,G)) . z = TRUE ) )

per cases ( (All (a,PA,G)) . z = TRUE or (All (b,PA,G)) . z = TRUE ) by A1, A2, BINARITH:3;

case A3: (All (a,PA,G)) . z = TRUE ; :: thesis: (All ((a 'or' b),PA,G)) . z = TRUE

for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(a 'or' b) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (a 'or' b) . x = TRUE )
assume A4: x in EqClass (z,(CompF (PA,G))) ; :: thesis: (a 'or' b) . x = TRUE
(a 'or' b) . x = (a . x) 'or' (b . x) by BVFUNC_1:def 4
.= TRUE 'or' (b . x) by A3, A4, BVFUNC_1:def 16
.= TRUE by BINARITH:10 ;
hence (a 'or' b) . x = TRUE ; :: thesis: verum

end;

hence (All ((a 'or' b),PA,G)) . z = TRUE by BVFUNC_1:def 16; :: thesis: verum

end;

case A5: (All (b,PA,G)) . z = TRUE ; :: thesis: (All ((a 'or' b),PA,G)) . z = TRUE

for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(a 'or' b) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (a 'or' b) . x = TRUE )
assume A6: x in EqClass (z,(CompF (PA,G))) ; :: thesis: (a 'or' b) . x = TRUE
(a 'or' b) . x = (a . x) 'or' (b . x) by BVFUNC_1:def 4
.= (a . x) 'or' TRUE by A5, A6, BVFUNC_1:def 16
.= TRUE by BINARITH:10 ;
hence (a 'or' b) . x = TRUE ; :: thesis: verum

end;

hence (All ((a 'or' b),PA,G)) . z = TRUE by BVFUNC_1:def 16; :: thesis: verum

end;

end;

end;

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