let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A holds QuantNbr (p '&' q) = (QuantNbr p) + (QuantNbr q)
let p, q be Element of CQC-WFF A; :: thesis: QuantNbr (p '&' q) = (QuantNbr p) + (QuantNbr q)
deffunc H₅( Element of CQC-WFF A) -> Element of NAT = QuantNbr $1;
A1: for p being Element of CQC-WFF A
for d being Element of NAT holds
( d = H₅(p) iff ex F being Function of (CQC-WFF A),NAT st
( d = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H₁(k,P,l) & F . ('not' r) = H₂(F . r) & F . (r '&' s) = H₃(F . r,F . s) & F . (All (x,r)) = H₄(x,F . r) ) ) ) ) by Def6;
thus H₅(p '&' q) = H₃(H₅(p),H₅(q)) from CQC_LANG:sch 8(A1); :: thesis: verum