let S be non empty set ; :: thesis: for R being total Relation of S,S
for BASSIGN being non empty Subset of (ModelSP S)
for G1, G2 being Subset of S st G1 c= G2 holds
for s being Element of S st s |= Tau (G1,R,BASSIGN) holds
s |= Tau (G2,R,BASSIGN)
let R be total Relation of S,S; :: thesis: for BASSIGN being non empty Subset of (ModelSP S)
for G1, G2 being Subset of S st G1 c= G2 holds
for s being Element of S st s |= Tau (G1,R,BASSIGN) holds
s |= Tau (G2,R,BASSIGN)
let BASSIGN be non empty Subset of (ModelSP S); :: thesis: for G1, G2 being Subset of S st G1 c= G2 holds
for s being Element of S st s |= Tau (G1,R,BASSIGN) holds
s |= Tau (G2,R,BASSIGN)
let G1, G2 be Subset of S; :: thesis: ( G1 c= G2 implies for s being Element of S st s |= Tau (G1,R,BASSIGN) holds
s |= Tau (G2,R,BASSIGN) )
set Tau1 = Tau (G1,R,BASSIGN);
set Tau2 = Tau (G2,R,BASSIGN);
assume A1: G1 c= G2 ; :: thesis: for s being Element of S st s |= Tau (G1,R,BASSIGN) holds
s |= Tau (G2,R,BASSIGN)
let s be Element of S; :: thesis: ( s |= Tau (G1,R,BASSIGN) implies s |= Tau (G2,R,BASSIGN) )
assume s |= Tau (G1,R,BASSIGN) ; :: thesis: s |= Tau (G2,R,BASSIGN)
then (Fid ((Tau (G1,R,BASSIGN)),S)) . s = TRUE ;
then (chi (G1,S)) . s = 1 by Def64;
then s in G1 by FUNCT_3:def 3;
then (chi (G2,S)) . s = 1 by A1, FUNCT_3:def 3;
then (Fid ((Tau (G2,R,BASSIGN)),S)) . s = TRUE by Def64;
hence s |= Tau (G2,R,BASSIGN) ; :: thesis: verum