let A be non empty set ; :: thesis: for J being interpretation of A
for v being Element of Valuations_in A
for S1, S2 being Element of CQC-Sub-WFF st S1 `2 = S2 `2 holds
( ( J,v . (Val_S v,S1) |= S1 & J,v . (Val_S v,S2) |= S2 ) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 )
let J be interpretation of A; :: thesis: for v being Element of Valuations_in A
for S1, S2 being Element of CQC-Sub-WFF st S1 `2 = S2 `2 holds
( ( J,v . (Val_S v,S1) |= S1 & J,v . (Val_S v,S2) |= S2 ) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 )
let v be Element of Valuations_in A; :: thesis: for S1, S2 being Element of CQC-Sub-WFF st S1 `2 = S2 `2 holds
( ( J,v . (Val_S v,S1) |= S1 & J,v . (Val_S v,S2) |= 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 implies ( ( J,v . (Val_S v,S1) |= S1 & J,v . (Val_S v,S2) |= S2 ) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 ) )
assume A1: S1 `2 = S2 `2 ; :: thesis: ( ( J,v . (Val_S v,S1) |= S1 & J,v . (Val_S v,S2) |= S2 ) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 )
then ( Val_S v,S1 = Val_S v,(CQCSub_& S1,S2) & Val_S v,S2 = Val_S v,(CQCSub_& S1,S2) ) by Th22;
then A2: ( ( J,v . (Val_S v,S1) |= S1 `1 & J,v . (Val_S v,S1) |= S2 `1 ) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= (S1 `1 ) '&' (S2 `1 ) ) by VALUAT_1:29;
( J,v . (Val_S v,(CQCSub_& S1,S2)) |= (S1 `1 ) '&' (S2 `1 ) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= (CQCSub_& S1,S2) `1 ) by A1, Th21;
hence ( ( J,v . (Val_S v,S1) |= S1 & J,v . (Val_S v,S2) |= S2 ) iff J,v . (Val_S v,(CQCSub_& S1,S2)) |= CQCSub_& S1,S2 ) by A1, A2, Def3; :: thesis: verum