consider F being Function of QC-WFF,F₁() such that
A3: F₈(F₃()) = F . F₃() and
A4: for p being Element of QC-WFF
for d1, d2 being Element of F₁() holds
( ( p = VERUM implies F . p = F₂() ) & ( p is atomic implies F . p = F₄(p) ) & ( p is negative & d1 = F . (the_argument_of p) implies F . p = F₅(d1) ) & ( p is conjunctive & d1 = F . (the_left_argument_of p) & d2 = F . (the_right_argument_of p) implies F . p = F₆(d1,d2) ) & ( p is universal & d1 = F . (the_scope_of p) implies F . p = F₇(p,d1) ) ) by A1;
F . (the_argument_of F₃()) = F₈((the_argument_of F₃())) by A1, A4;
hence F₈(F₃()) = F₅(F₈((the_argument_of F₃()))) by A2, A3, A4; :: thesis: verum