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 Th20:
  p is conjunctive implies p.x = ((the_left_argument_of p).x) '&'
  ((the_right_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 F2 being Function of QC-WFF(A),QC-WFF(A) such that
A3: (the_right_argument_of p).x = F2.(the_right_argument_of p) and
A4: for q holds F2.VERUM(A) = VERUM(A) & (q is atomic implies F2.q = (
  the_pred_symbol_of q)!Subst(the_arguments_of q,(A)a.0.-->x)) & (q is negative
implies F2.q = 'not' (F2.the_argument_of q) ) & (q is conjunctive implies F2.q
  = (F2.the_left_argument_of q) '&' (F2.the_right_argument_of q)) & (q is
universal implies F2.q = IFEQ(bound_in q,x,q,All(bound_in q,F2.the_scope_of q))
  ) by Def3;
A5: F2 = F by A2,A4,Lm2;
  consider F1 being Function of QC-WFF(A),QC-WFF(A) such that
A6: (the_left_argument_of p).x = F1.(the_left_argument_of p) and
A7: for q holds F1.VERUM(A) = VERUM(A) & (q is atomic implies F1.q = (
  the_pred_symbol_of q)!Subst(the_arguments_of q,(A)a.0.-->x)) & (q is negative
implies F1.q = 'not' (F1.the_argument_of q) ) & (q is conjunctive implies F1.q
  = (F1.the_left_argument_of q) '&' (F1.the_right_argument_of q)) & (q is
universal implies F1.q = IFEQ(bound_in q,x,q,All(bound_in q,F1.the_scope_of q))
  ) by Def3;
  F1 = F by A2,A7,Lm2;
  hence thesis by A1,A2,A6,A3,A5;
end;
