reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th25:
  for n being non empty natural number
  for X being non empty set
  for S being non empty non void n PC-correct QC-correct
  QCLangSignature over X
  for Q being language MSAlgebra over S holds
  {} in Args(In((the connectives of S).(n+5), the carrier' of S),Q) &
  for A being Formula of Q holds
  <*A*> in Args(In((the connectives of S).n, the carrier' of S),Q) &
  for B being Formula of Q holds
  (<*A,B*> in Args(In((the connectives of S).(n+1), the carrier' of S),Q)
  & ... &
  <*A,B*> in Args(In((the connectives of S).(n+4), the carrier' of S),Q)) &
  for x being Element of X holds
  <*A*> in Args(In((the quantifiers of S).(1,x), the carrier' of S),Q) &
  <*A*> in Args(In((the quantifiers of S).(2,x), the carrier' of S),Q)
  proof
    let n be non empty natural number;
    let X be non empty set;
    let S be non empty non void n PC-correct QC-correct
    QCLangSignature over X;
    let Q be language MSAlgebra over S;
    set f = the formula-sort of S;
A1: len the connectives of S >= n+5 by Def4;
    n > 0;
    then
A2: n >= 0+1 by NAT_1:13;
    n+0 <= n+5 & ... & n+4 <= n+5 by XREAL_1:6;
    then (1 <= n+0 & ... & 1 <= n+5) &
    (n+0 <= len the connectives of S & ... &
    n+5 <= len the connectives of S)
    by A1,A2,NAT_1:12,XXREAL_0:2; then
A3: n+0 in dom the connectives of S & ... & n+5 in dom the connectives of S
    by FINSEQ_3:25;
A4: (the connectives of S).(n+0) is_of_type <*f*>, f &
    ((the connectives of S).(n+1) is_of_type <*f,f*>, f & ... &
    (the connectives of S).(n+4) is_of_type <*f,f*>, f) &
    (the connectives of S).(n+5) is_of_type {}, f by Def4;
    In((the connectives of S).(n+5), the carrier' of S) is_of_type {},f
    by A4,A3,FUNCT_1:102,SUBSET_1:def 8;
    hence {} in Args(In((the connectives of S).(n+5), the carrier' of S),Q)
    by Th4;
    let A be Formula of Q;
    In((the connectives of S).(n+0), the carrier' of S) is_of_type <*f*>,f
    by A4,A3,FUNCT_1:102,SUBSET_1:def 8;
    hence <*A*> in Args(In((the connectives of S).n, the carrier' of S),Q)
    by Th5;
    let B be Formula of Q;
    In((the connectives of S).(n+1), the carrier' of S)
    = (the connectives of S).(n+1) & ... &
    In((the connectives of S).(n+4), the carrier' of S)
    = (the connectives of S).(n+4) by A3,FUNCT_1:102,SUBSET_1:def 8;
    then
    In((the connectives of S).(n+1), the carrier' of S) is_of_type <*f,f*>, f
    & ... &
    In((the connectives of S).(n+4), the carrier' of S) is_of_type <*f,f*>, f
    by Def4;
    hence <*A,B*> in Args(In((the connectives of S).(n+1),
    the carrier' of S),Q) & ... &
    <*A,B*> in Args(In((the connectives of S).(n+4), the carrier' of S),Q)
    by Th6;
    let x be Element of X;
    the quant-sort of S = {1,2} by Def5;
    then
A5: 1 in the quant-sort of S & 2 in the quant-sort of S by TARSKI:def 2;
    then [1,x] in [:the quant-sort of S,X:] &
    dom the quantifiers of S = [:the quant-sort of S, X:] &
    [2,x] in [:the quant-sort of S,X:] by FUNCT_2:def 1,ZFMISC_1:def 2;
    then In((the quantifiers of S).(1,x), the carrier' of S)
    = (the quantifiers of S).(1,x) &
    In((the quantifiers of S).(2,x), the carrier' of S)
    = (the quantifiers of S).(2,x) by SUBSET_1:def 8,FUNCT_1:102;
    then In((the quantifiers of S).(1,x), the carrier' of S) is_of_type <*f*>,f
    & In((the quantifiers of S).(2,x), the carrier' of S) is_of_type <*f*>,f
    by A5,Def5;
    hence <*A*> in Args(In((the quantifiers of S).(1,x), the carrier' of S),Q)
    & <*A*> in Args(In((the quantifiers of S).(2,x), the carrier' of S),Q)
    by Th5;
  end;
