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 that
A1: S1 `2 = S2 `2 and
A2: ( ( 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) )
A3: ( 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 A2;
( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v |= (CQC_Sub S1) '&' (CQC_Sub S2) ) by A1, Th24;
hence ( J,v |= CQC_Sub (CQCSub_& (S1,S2)) iff J,v . (Val_S (v,(CQCSub_& (S1,S2)))) |= CQCSub_& (S1,S2) ) by A1, A3, Th25, VALUAT_1:18; :: thesis: verum