let S be Element of QC-Sub-WFF ; :: thesis: ( S is Sub_VERUM implies CQC_Sub S is Element of CQC-WFF )
assume A1: S is Sub_VERUM ; :: thesis: CQC_Sub S is Element of CQC-WFF
consider F being Function of QC-Sub-WFF ,QC-WFF such that
A2: ( CQC_Sub S = F . S & ( for S' being Element of QC-Sub-WFF holds
( ( S' is Sub_VERUM implies F . S' = VERUM ) & ( S' is Sub_atomic implies F . S' = (the_pred_symbol_of (S' `1 )) ! (CQC_Subst (Sub_the_arguments_of S'),(S' `2 )) ) & ( S' is Sub_negative implies F . S' = 'not' (F . (Sub_the_argument_of S')) ) & ( S' is Sub_conjunctive implies F . S' = (F . (Sub_the_left_argument_of S')) '&' (F . (Sub_the_right_argument_of S')) ) & ( S' is Sub_universal implies F . S' = Quant S',(F . (Sub_the_scope_of S')) ) ) ) ) by Def38;
thus CQC_Sub S is Element of CQC-WFF by A1, A2; :: thesis: verum