let Al be QC-alphabet ; :: thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S1, S2 being Element of CQC-Sub-WFF Al 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 A be non empty set ; :: thesis: for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S1, S2 being Element of CQC-Sub-WFF Al 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 Al,A; :: thesis: for v being Element of Valuations_in (Al,A)
for S1, S2 being Element of CQC-Sub-WFF Al 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 (Al,A); :: thesis: for S1, S2 being Element of CQC-Sub-WFF Al 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 Al; :: 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))) by Th21;
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_1:18;
( 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, Th20;
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; :: thesis: verum