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 |= EX f iff ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= 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 |= EX f iff ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= 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 |= EX f iff ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= f ) )
let BASSIGN be non empty Subset of (ModelSP S); :: thesis: for f being Assign of (BASSModel (R,BASSIGN)) holds
( s |= EX f iff ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= f ) )
let f be Assign of (BASSModel (R,BASSIGN)); :: thesis: ( s |= EX f iff ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= f ) )
A1: EX f = EneXt_0 (f,R) by Def46;
A2: ( ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= f ) implies s |= EX f )

proof

assume A3: ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= f ) ; :: thesis: s |= EX f
ex pai being inf_path of R st
( pai . 0 = s & (Fid (f,S)) . (pai . 1) = TRUE ) by A3;
then EneXt_univ (s,(Fid (f,S)),R) = TRUE by Def44;
then (Fid ((EX f),S)) . s = TRUE by A1, Def45;
hence s |= EX f ; :: thesis: verum

end;

( s |= EX f implies ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= f ) )

proof

assume s |= EX f ; :: thesis: ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= f )
then (Fid ((EneXt_0 (f,R)),S)) . s = TRUE by A1;
then EneXt_univ (s,(Fid (f,S)),R) = TRUE by Def45;
then consider pai being inf_path of R such that
A4: pai . 0 = s and
A5: (Fid (f,S)) . (pai . 1) = TRUE by Def44;
take pai ; :: thesis: ( pai . 0 = s & pai . 1 |= f )
thus ( pai . 0 = s & pai . 1 |= f ) by A4, A5; :: thesis: verum

end;

hence ( s |= EX f iff ex pai being inf_path of R st
( pai . 0 = s & pai . 1 |= f ) ) by A2; :: thesis: verum