let A be QC-alphabet ; :: thesis: for S being Element of QC-Sub-WFF A st S is A -Sub_VERUM holds
CQC_Sub S is Element of CQC-WFF A
let S be Element of QC-Sub-WFF A; :: thesis: ( S is A -Sub_VERUM implies CQC_Sub S is Element of CQC-WFF A )
assume A1: S is A -Sub_VERUM ; :: thesis: CQC_Sub S is Element of CQC-WFF A
ex F being Function of (QC-Sub-WFF A),(QC-WFF A) st
( CQC_Sub S = F . S & ( for S9 being Element of QC-Sub-WFF A holds
( ( S9 is A -Sub_VERUM implies F . S9 = VERUM A ) & ( S9 is Sub_atomic implies F . S9 = (the_pred_symbol_of (S9 `1)) ! (CQC_Subst ((Sub_the_arguments_of S9),(S9 `2))) ) & ( S9 is Sub_negative implies F . S9 = 'not' (F . (Sub_the_argument_of S9)) ) & ( S9 is Sub_conjunctive implies F . S9 = (F . (Sub_the_left_argument_of S9)) '&' (F . (Sub_the_right_argument_of S9)) ) & ( S9 is Sub_universal implies F . S9 = Quant (S9,(F . (Sub_the_scope_of S9))) ) ) ) ) by Def38;
hence CQC_Sub S is Element of CQC-WFF A by A1; :: thesis: verum