let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, u being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds (Ex a,PA,G) 'imp' u '<' Ex (a 'imp' u),PA,G
let G be Subset of (PARTITIONS Y); :: thesis: for a, u being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds (Ex a,PA,G) 'imp' u '<' Ex (a 'imp' u),PA,G
let a, u be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds (Ex a,PA,G) 'imp' u '<' Ex (a 'imp' u),PA,G
let PA be a_partition of Y; :: thesis: (Ex a,PA,G) 'imp' u '<' Ex (a 'imp' u),PA,G
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((Ex a,PA,G) 'imp' u) . z = TRUE or (Ex (a 'imp' u),PA,G) . z = TRUE )
A1: z in EqClass z,(CompF PA,G) by EQREL_1:def 8;
assume ((Ex a,PA,G) 'imp' u) . z = TRUE ; :: thesis: (Ex (a 'imp' u),PA,G) . z = TRUE
then A2: ('not' ((Ex a,PA,G) . z)) 'or' (u . z) = TRUE by BVFUNC_1:def 11;
A3: ( u . z = TRUE or u . z = FALSE ) by XBOOLEAN:def 3;

now

per cases ( 'not' ((Ex a,PA,G) . z) = TRUE or u . z = TRUE ) by A2, A3, BINARITH:7;

case 'not' ((Ex a,PA,G) . z) = TRUE ; :: thesis: (Ex (a 'imp' u),PA,G) . z = TRUE

then (Ex a,PA,G) . z = FALSE by MARGREL1:41;
then A4: a . z <> TRUE by A1, BVFUNC_1:def 20;
(a 'imp' u) . z = ('not' (a . z)) 'or' (u . z) by BVFUNC_1:def 11
.= ('not' FALSE ) 'or' (u . z) by A4, XBOOLEAN:def 3
.= TRUE 'or' (u . z) by MARGREL1:41
.= TRUE by BINARITH:19 ;
hence (Ex (a 'imp' u),PA,G) . z = TRUE by A1, BVFUNC_1:def 20; :: thesis: verum

end;

case A5: u . z = TRUE ; :: thesis: (Ex (a 'imp' u),PA,G) . z = TRUE

(a 'imp' u) . z = ('not' (a . z)) 'or' (u . z) by BVFUNC_1:def 11
.= TRUE by A5, BINARITH:19 ;
hence (Ex (a 'imp' u),PA,G) . z = TRUE by A1, BVFUNC_1:def 20; :: thesis: verum

end;

end;

end;

hence (Ex (a 'imp' u),PA,G) . z = TRUE ; :: thesis: verum