let A be non empty set ; :: thesis: for J being interpretation of A
for S1, S2 being Element of CQC-Sub-WFF st S1 `2 = S2 `2 & ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S v,S1) |= S1 ) ) & ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S v,S2) |= S2 ) ) holds
for v being Element of Valuations_in A holds
( J,v |= CQC_Sub (CQCSub_& S1,S2) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 )
let J be interpretation of A; :: thesis: for S1, S2 being Element of CQC-Sub-WFF st S1 `2 = S2 `2 & ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S v,S1) |= S1 ) ) & ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S v,S2) |= S2 ) ) holds
for v being Element of Valuations_in A holds
( J,v |= CQC_Sub (CQCSub_& S1,S2) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 )
let S1, S2 be Element of CQC-Sub-WFF ; :: thesis: ( S1 `2 = S2 `2 & ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S v,S1) |= S1 ) ) & ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S v,S2) |= S2 ) ) implies for v being Element of Valuations_in A holds
( J,v |= CQC_Sub (CQCSub_& S1,S2) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 ) )
assume A1: ( S1 `2 = S2 `2 & ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S1 iff J,v . (Val_S v,S1) |= S1 ) ) & ( for v being Element of Valuations_in A holds
( J,v |= CQC_Sub S2 iff J,v . (Val_S v,S2) |= S2 ) ) ) ; :: thesis: for v being Element of Valuations_in A holds
( J,v |= CQC_Sub (CQCSub_& S1,S2) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 )
let v be Element of Valuations_in A; :: thesis: ( J,v |= CQC_Sub (CQCSub_& S1,S2) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 )
A2: ( J,v |= CQC_Sub (CQCSub_& S1,S2) iff J,v |= (CQC_Sub S1) '&' (CQC_Sub S2) ) by A1, Th24;
( J,v |= CQC_Sub S1 & J,v |= CQC_Sub S2 iff ( J,v . (Val_S v,S1) |= S1 & J,v . (Val_S v,S2) |= S2 ) ) by A1;
hence ( J,v |= CQC_Sub (CQCSub_& S1,S2) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 ) by A1, A2, Th25, VALUAT_1:29; :: thesis: verum