let S be non empty set ; :: thesis: for BASSIGN being non empty Subset of (ModelSP (Inf_seq S))
for t being Element of Inf_seq S
for f being Assign of (LTLModel S,BASSIGN) holds
( t |= 'not' f iff t |/= f )
let BASSIGN be non empty Subset of (ModelSP (Inf_seq S)); :: thesis: for t being Element of Inf_seq S
for f being Assign of (LTLModel S,BASSIGN) holds
( t |= 'not' f iff t |/= f )
let t be Element of Inf_seq S; :: thesis: for f being Assign of (LTLModel S,BASSIGN) holds
( t |= 'not' f iff t |/= f )
let f be Assign of (LTLModel S,BASSIGN); :: thesis: ( t |= 'not' f iff t |/= f )
set V = LTLModel S,BASSIGN;
set S1 = Inf_seq S;
A1: 'not' f = Not_0 f,S by Def43;
thus ( t |= 'not' f implies t |/= f ) :: thesis: ( t |/= f implies t |= 'not' f )

proof

assume t |= 'not' f ; :: thesis: t |/= f
then (Fid ('not' f),(Inf_seq S)) . t = TRUE by Def59;
then 'not' (Castboolean ((Fid f,(Inf_seq S)) . t)) = TRUE by A1, Def42;
then (Fid f,(Inf_seq S)) . t = FALSE by MODELC_1:def 4;
hence t |/= f by Def59; :: thesis: verum

end;

assume t |/= f ; :: thesis: t |= 'not' f
then not (Fid f,(Inf_seq S)) . t = TRUE by Def59;
then not Castboolean ((Fid f,(Inf_seq S)) . t) = TRUE by MODELC_1:def 4;
then Castboolean ((Fid f,(Inf_seq S)) . t) = FALSE by XBOOLEAN:def 3;
then 'not' (Castboolean ((Fid f,(Inf_seq S)) . t)) = TRUE ;
then (Fid ('not' f),(Inf_seq S)) . t = TRUE by Def42, A1;
hence t |= 'not' f by Def59; :: thesis: verum