let Al be QC-alphabet ; :: thesis: for A being non empty set
for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let A be non empty set ; :: thesis: for J being interpretation of Al,A
for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let J be interpretation of Al,A; :: thesis: for S being Element of CQC-Sub-WFF Al st S is Al -Sub_VERUM holds
for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let S be Element of CQC-Sub-WFF Al; :: thesis: ( S is Al -Sub_VERUM implies for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S ) )
assume A1: S is Al -Sub_VERUM ; :: thesis: for v being Element of Valuations_in (Al,A) holds
( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
let v be Element of Valuations_in (Al,A); :: thesis: ( J,v |= CQC_Sub S iff J,v . (Val_S (v,S)) |= S )
ex Sub being CQC_Substitution of Al st S = [(VERUM Al),Sub] by A1, SUBSTUT1:def 19;
then S `1 = VERUM Al ;
then ( J,v . (Val_S (v,S)) |= VERUM Al iff J,v . (Val_S (v,S)) |= S ) ;
hence ( J,v |= CQC_Sub S implies J,v . (Val_S (v,S)) |= S ) by VALUAT_1:32; :: thesis: ( J,v . (Val_S (v,S)) |= S implies J,v |= CQC_Sub S )
( J,v . (Val_S (v,S)) |= S implies J,v |= VERUM Al ) by VALUAT_1:32;
hence ( J,v . (Val_S (v,S)) |= S implies J,v |= CQC_Sub S ) by A1, Th3; :: thesis: verum