reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;

theorem Th6:
  len fin <= len (fin^fin1) & len fin1 <= len (fin^fin1) & (fin <>
  {} implies 1 <= len fin & len fin1 < len (fin1^fin))
proof
  len (fin^fin1) = len fin + len fin1 by FINSEQ_1:22;
  hence len fin <= len (fin^fin1) & len fin1 <= len (fin^fin1) by NAT_1:12;
  assume fin <> {};
  then
A1: 0+1 <= len fin by NAT_1:13;
  then len fin1 + 1 <= len fin + len fin1 by XREAL_1:6;
  then len fin1 + 1 <= len (fin1^fin) by FINSEQ_1:22;
  hence thesis by A1,NAT_1:13;
end;
