let Y be non empty set ; :: thesis: for G being Subset of ()
for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All ((),PA,G)) 'or' (All ((),PA,G))

let G be Subset of (); :: thesis: for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All ((),PA,G)) 'or' (All ((),PA,G))

let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All ((),PA,G)) 'or' (All ((),PA,G))
let PA be a_partition of Y; :: thesis: 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All ((),PA,G)) 'or' (All ((),PA,G))
A1: All ((),PA,G) = B_INF ((),(CompF (PA,G))) by BVFUNC_2:def 9;
A2: Ex (b,PA,G) = B_SUP (b,(CompF (PA,G))) by BVFUNC_2:def 10;
A3: Ex (a,PA,G) = B_SUP (a,(CompF (PA,G))) by BVFUNC_2:def 10;
A4: (All ((),PA,G)) 'or' (All ((),PA,G)) '<' 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))
proof
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((All ((),PA,G)) 'or' (All ((),PA,G))) . z = TRUE or ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE )
A5: ( (All ((),PA,G)) . z = TRUE or (All ((),PA,G)) . z = FALSE ) by XBOOLEAN:def 3;
assume ((All ((),PA,G)) 'or' (All ((),PA,G))) . z = TRUE ; :: thesis: ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE
then A6: ((All ((),PA,G)) . z) 'or' ((All ((),PA,G)) . z) = TRUE by BVFUNC_1:def 4;
per cases ( (All ((),PA,G)) . z = TRUE or (All ((),PA,G)) . z = TRUE ) by ;
suppose A7: (All ((),PA,G)) . z = TRUE ; :: thesis: ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE
A8: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
() . x = TRUE
assume ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not () . x = TRUE ) ; :: thesis: contradiction
then (B_INF ((),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
hence contradiction by A7, BVFUNC_2:def 9; :: thesis: verum
end;
A9: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x <> TRUE
let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies a . x <> TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: a . x <> TRUE
then ('not' a) . x = TRUE by A8;
then 'not' (a . x) = TRUE by MARGREL1:def 19;
hence a . x <> TRUE by MARGREL1:11; :: thesis: verum
end;
thus ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = 'not' (((Ex (a,PA,G)) '&' (Ex (b,PA,G))) . z) by MARGREL1:def 19
.= 'not' (((Ex (a,PA,G)) . z) '&' ((Ex (b,PA,G)) . z)) by MARGREL1:def 20
.= 'not' (FALSE '&' ((Ex (b,PA,G)) . z)) by
.= 'not' FALSE by MARGREL1:12
.= TRUE by MARGREL1:11 ; :: thesis: verum
end;
suppose A10: (All ((),PA,G)) . z = TRUE ; :: thesis: ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE
A11: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
() . x = TRUE
assume ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not () . x = TRUE ) ; :: thesis: contradiction
then (B_INF ((),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
hence contradiction by A10, BVFUNC_2:def 9; :: thesis: verum
end;
A12: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x <> TRUE
let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies b . x <> TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: b . x <> TRUE
then ('not' b) . x = TRUE by A11;
then 'not' (b . x) = TRUE by MARGREL1:def 19;
hence b . x <> TRUE by MARGREL1:11; :: thesis: verum
end;
thus ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = 'not' (((Ex (a,PA,G)) '&' (Ex (b,PA,G))) . z) by MARGREL1:def 19
.= 'not' (((Ex (a,PA,G)) . z) '&' ((Ex (b,PA,G)) . z)) by MARGREL1:def 20
.= 'not' (((Ex (a,PA,G)) . z) '&' FALSE) by
.= 'not' FALSE by MARGREL1:12
.= TRUE by MARGREL1:11 ; :: thesis: verum
end;
end;
end;
A13: All ((),PA,G) = B_INF ((),(CompF (PA,G))) by BVFUNC_2:def 9;
'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) '<' (All ((),PA,G)) 'or' (All ((),PA,G))
proof
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE or ((All ((),PA,G)) 'or' (All ((),PA,G))) . z = TRUE )
assume ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE ; :: thesis: ((All ((),PA,G)) 'or' (All ((),PA,G))) . z = TRUE
then 'not' (((Ex (a,PA,G)) '&' (Ex (b,PA,G))) . z) = TRUE by MARGREL1:def 19;
then ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) . z = FALSE by MARGREL1:11;
then A14: ((Ex (a,PA,G)) . z) '&' ((Ex (b,PA,G)) . z) = FALSE by MARGREL1:def 20;
per cases ( (Ex (a,PA,G)) . z = FALSE or (Ex (b,PA,G)) . z = FALSE ) by ;
suppose A15: (Ex (a,PA,G)) . z = FALSE ; :: thesis: ((All ((),PA,G)) 'or' (All ((),PA,G))) . z = TRUE
A16: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
() . x = TRUE
let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies () . x = TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: () . x = TRUE
then a . x <> TRUE by ;
then a . x = FALSE by XBOOLEAN:def 3;
then 'not' (a . x) = TRUE by MARGREL1:11;
hence ('not' a) . x = TRUE by MARGREL1:def 19; :: thesis: verum
end;
thus ((All ((),PA,G)) 'or' (All ((),PA,G))) . z = ((All ((),PA,G)) . z) 'or' ((All ((),PA,G)) . z) by BVFUNC_1:def 4
.= TRUE 'or' ((All ((),PA,G)) . z) by
.= TRUE by BINARITH:10 ; :: thesis: verum
end;
suppose A17: (Ex (b,PA,G)) . z = FALSE ; :: thesis: ((All ((),PA,G)) 'or' (All ((),PA,G))) . z = TRUE
A18: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
() . x = TRUE
let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies () . x = TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: () . x = TRUE
then b . x <> TRUE by ;
then b . x = FALSE by XBOOLEAN:def 3;
then 'not' (b . x) = TRUE by MARGREL1:11;
hence ('not' b) . x = TRUE by MARGREL1:def 19; :: thesis: verum
end;
thus ((All ((),PA,G)) 'or' (All ((),PA,G))) . z = ((All ((),PA,G)) . z) 'or' ((All ((),PA,G)) . z) by BVFUNC_1:def 4
.= ((All ((),PA,G)) . z) 'or' TRUE by
.= TRUE by BINARITH:10 ; :: thesis: verum
end;
end;
end;
hence 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All ((),PA,G)) 'or' (All ((),PA,G)) by ; :: thesis: verum