reserve A for QC-alphabet;
reserve a,b,b1,b2,c,d for object,
  i,j,k,n for Nat,
  x,y,x1,x2 for bound_QC-variable of A,
  P for QC-pred_symbol of k,A,
  ll for CQC-variable_list of k,A,
  l1 ,l2 for FinSequence of QC-variables(A),
  p for QC-formula of A,
  s,t for QC-symbol of A;
reserve Sub for CQC_Substitution of A;
reserve finSub for finite CQC_Substitution of A;
reserve e for Element of vSUB(A);
reserve S,S9,S1,S2,S19,S29,T1,T2 for Element of QC-Sub-WFF(A);
reserve B for Element of [:QC-Sub-WFF(A),bound_QC-variables(A):];
reserve SQ for second_Q_comp of B;

theorem Th23:
  S is Sub_conjunctive implies len @((Sub_the_left_argument_of(S))
  `1) < len @(S`1) & len @((Sub_the_right_argument_of(S))`1) < len @(S`1)
proof
  assume S is Sub_conjunctive;
  then consider S1,S2 such that
A1: S = Sub_&(S1,S2) & S1`2 = S2`2;
  S = [(S1`1) '&' (S2`1),S1`2] by A1,Def21;
  then
A2: S`1 = (S1`1) '&' (S2`1);
  (S1`1) '&' (S2`1) is conjunctive;
  then
A3: len @the_left_argument_of (S1`1) '&' (S2`1) < len @(S`1) & len @
  the_right_argument_of (S1`1) '&' (S2`1) < len @(S`1) by A2,QC_LANG1:15;
  (Sub_the_right_argument_of(S))`1 = S2`1 & (Sub_the_left_argument_of(S))
  `1 = S1`1 by A1,Th18,Th19;
  hence thesis by A3,QC_LANG2:4;
end;
