let S be non empty set ; :: thesis: for R being total Relation of S,S
for s being Element of S
for BASSIGN being non empty Subset of (ModelSP S)
for f being Assign of (BASSModel (R,BASSIGN)) holds
( s |= EG f iff ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) )
let R be total Relation of S,S; :: thesis: for s being Element of S
for BASSIGN being non empty Subset of (ModelSP S)
for f being Assign of (BASSModel (R,BASSIGN)) holds
( s |= EG f iff ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) )
let s be Element of S; :: thesis: for BASSIGN being non empty Subset of (ModelSP S)
for f being Assign of (BASSModel (R,BASSIGN)) holds
( s |= EG f iff ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) )
let BASSIGN be non empty Subset of (ModelSP S); :: thesis: for f being Assign of (BASSModel (R,BASSIGN)) holds
( s |= EG f iff ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) )
let f be Assign of (BASSModel (R,BASSIGN)); :: thesis: ( s |= EG f iff ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) )
A1: EG f = EGlobal_0 (f,R) by Def49;
A2: ( ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) implies s |= EG f )

proof

assume A3: ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) ; :: thesis: s |= EG f
ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds (Fid (f,S)) . (pai . n) = TRUE ) )

proof

consider pai being inf_path of R such that
A4: pai . 0 = s and
A5: for n being Element of NAT holds pai . n |= f by A3;
take pai ; :: thesis: ( pai . 0 = s & ( for n being Element of NAT holds (Fid (f,S)) . (pai . n) = TRUE ) )
for n being Element of NAT holds (Fid (f,S)) . (pai . n) = TRUE by A5, Def59;
hence ( pai . 0 = s & ( for n being Element of NAT holds (Fid (f,S)) . (pai . n) = TRUE ) ) by A4; :: thesis: verum

end;

then EGlobal_univ (s,(Fid (f,S)),R) = TRUE by Def47;
then (Fid ((EG f),S)) . s = TRUE by A1, Def48;
hence s |= EG f ; :: thesis: verum

end;

( s |= EG f implies ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) )

proof

assume s |= EG f ; :: thesis: ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) )
then (Fid ((EGlobal_0 (f,R)),S)) . s = TRUE by A1;
then EGlobal_univ (s,(Fid (f,S)),R) = TRUE by Def48;
then consider pai being inf_path of R such that
A6: pai . 0 = s and
A7: for n being Element of NAT holds (Fid (f,S)) . (pai . n) = TRUE by Def47;
take pai ; :: thesis: ( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) )
for n being Element of NAT holds pai . n |= f by A7;
hence ( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) by A6; :: thesis: verum

end;

hence ( s |= EG f iff ex pai being inf_path of R st
( pai . 0 = s & ( for n being Element of NAT holds pai . n |= f ) ) ) by A2; :: thesis: verum