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 All ((a 'imp' b),PA,G) = All ((() 'or' b),PA,G)

let G be Subset of (); :: thesis: for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds All ((a 'imp' b),PA,G) = All ((() 'or' b),PA,G)

let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All ((a 'imp' b),PA,G) = All ((() 'or' b),PA,G)
let PA be a_partition of Y; :: thesis: All ((a 'imp' b),PA,G) = All ((() 'or' b),PA,G)
A1: All ((() 'or' b),PA,G) '<' All ((a 'imp' b),PA,G)
proof
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((() 'or' b),PA,G)) . z = TRUE or (All ((a 'imp' b),PA,G)) . z = TRUE )
assume A2: (All ((() 'or' b),PA,G)) . z = TRUE ; :: thesis: (All ((a 'imp' b),PA,G)) . z = TRUE
A3: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(() 'or' b) . x = TRUE
assume ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not (() 'or' b) . x = TRUE ) ; :: thesis: contradiction
then (B_INF ((() 'or' b),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
hence contradiction by A2, BVFUNC_2:def 9; :: thesis: verum
end;
for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(a 'imp' b) . x = TRUE
proof
let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (a 'imp' b) . x = TRUE )
A4: ( ('not' a) . x = TRUE or () . x = FALSE ) by XBOOLEAN:def 3;
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: (a 'imp' b) . x = TRUE
then (('not' a) 'or' b) . x = TRUE by A3;
then A5: (('not' a) . x) 'or' (b . x) = TRUE by BVFUNC_1:def 4;
per cases ) . x = TRUE or b . x = TRUE ) by ;
suppose ('not' a) . x = TRUE ; :: thesis: (a 'imp' b) . x = TRUE
then 'not' (a . x) = TRUE by MARGREL1:def 19;
hence (a 'imp' b) . x = TRUE 'or' (b . x) by BVFUNC_1:def 8
.= TRUE by BINARITH:10 ;
:: thesis: verum
end;
suppose b . x = TRUE ; :: thesis: (a 'imp' b) . x = TRUE
hence (a 'imp' b) . x = ('not' (a . x)) 'or' TRUE by BVFUNC_1:def 8
.= TRUE by BINARITH:10 ;
:: thesis: verum
end;
end;
end;
then (B_INF ((a 'imp' b),(CompF (PA,G)))) . z = TRUE by BVFUNC_1:def 16;
hence (All ((a 'imp' b),PA,G)) . z = TRUE by BVFUNC_2:def 9; :: thesis: verum
end;
All ((a 'imp' b),PA,G) '<' All ((() 'or' b),PA,G)
proof
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a 'imp' b),PA,G)) . z = TRUE or (All ((() 'or' b),PA,G)) . z = TRUE )
assume A6: (All ((a 'imp' b),PA,G)) . z = TRUE ; :: thesis: (All ((() 'or' b),PA,G)) . z = TRUE
A7: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(a 'imp' b) . x = TRUE
assume ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not (a 'imp' b) . x = TRUE ) ; :: thesis: contradiction
then (B_INF ((a 'imp' b),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
hence contradiction by A6, BVFUNC_2:def 9; :: thesis: verum
end;
now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(() 'or' b) . x = TRUE
let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (() 'or' b) . b1 = TRUE )
A8: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: (() 'or' b) . b1 = TRUE
then (a 'imp' b) . x = TRUE by A7;
then A9: ('not' (a . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 8;
per cases ( 'not' (a . x) = TRUE or b . x = TRUE ) by ;
suppose 'not' (a . x) = TRUE ; :: thesis: (() 'or' b) . b1 = TRUE
then ('not' a) . x = TRUE by MARGREL1:def 19;
hence (() 'or' b) . x = TRUE 'or' (b . x) by BVFUNC_1:def 4
.= TRUE by BINARITH:10 ;
:: thesis: verum
end;
suppose b . x = TRUE ; :: thesis: (() 'or' b) . b1 = TRUE
hence (() 'or' b) . x = (() . x) 'or' TRUE by BVFUNC_1:def 4
.= TRUE by BINARITH:10 ;
:: thesis: verum
end;
end;
end;
then (B_INF ((() 'or' b),(CompF (PA,G)))) . z = TRUE by BVFUNC_1:def 16;
hence (All ((() 'or' b),PA,G)) . z = TRUE by BVFUNC_2:def 9; :: thesis: verum
end;
hence All ((a 'imp' b),PA,G) = All ((() 'or' b),PA,G) by ; :: thesis: verum