reserve A for QC-alphabet;
reserve i,j,k for Nat;
reserve f for Substitution of A;
reserve x,y for bound_QC-variable of A;
reserve a for free_QC-variable of A;
reserve p,q for Element of QC-WFF(A);
reserve l,l1,l2,ll for FinSequence of QC-variables(A);
reserve r,s for Element of CQC-WFF(A);

theorem Th18:
  p is negative implies p.x = 'not'((the_argument_of p).x)
proof
  consider F being Function of QC-WFF(A),QC-WFF(A) such that
A1: p.x = F.p and
A2: for q holds F.VERUM(A) = VERUM(A) & (q is atomic implies F.q = (
  the_pred_symbol_of q)!Subst(the_arguments_of q,(A)a.0.-->x)) & (q is negative
implies F.q = 'not' (F.the_argument_of q) ) & (q is conjunctive implies F.q = (
  F.the_left_argument_of q) '&' (F.the_right_argument_of q)) & (q is universal
  implies F.q = IFEQ(bound_in q,x,q,All(bound_in q,F.the_scope_of q))) by Def3;
  consider G being Function of QC-WFF(A),QC-WFF(A) such that
A3: (the_argument_of p).x = G.(the_argument_of p) and
A4: for q holds G.VERUM(A) = VERUM(A) & (q is atomic implies G.q = (
  the_pred_symbol_of q)!Subst(the_arguments_of q,(A)a.0.-->x)) & (q is negative
implies G.q = 'not' (G.the_argument_of q) ) & (q is conjunctive implies G.q = (
  G.the_left_argument_of q) '&' (G.the_right_argument_of q)) & (q is universal
  implies G.q = IFEQ(bound_in q,x,q,All(bound_in q,G.the_scope_of q))) by Def3;
  F = G by A2,A4,Lm2;
  hence thesis by A1,A2,A3;
end;
