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;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];

theorem Th17:
  len f > 0 implies (J,v |= Ant(f) & J,v |= Suc(f) iff J,v |= f)
proof
  assume
A1: len f > 0;
  thus J,v |= Ant(f) & J,v |= Suc(f) implies J,v |= f
  proof
    assume that
A2: J,v |= Ant(f) and
A3: J,v |= Suc(f);
    let p;
    assume p in rng(f);
    then p in rng(Ant(f)) \/ {Suc(f)} by A1,Th3;
    then
A4: p in rng(Ant(f)) or p in {Suc(f)} by XBOOLE_0:def 3;
    J,v |= rng(Ant(f)) by A2;
    hence thesis by A3,A4,TARSKI:def 1;
  end;
  thus J,v |= f implies J,v |= Ant(f) & J,v |= Suc(f)
  proof
    assume
A5: J,v |= rng(f);
    thus J,v |= rng(Ant(f))
    proof
A6:   rng(Ant(f)) c= rng(Ant(f)) \/ {Suc(f)} by XBOOLE_1:7;
      let p;
      assume p in rng(Ant(f));
      then p in rng(Ant(f)) \/ {Suc(f)} by A6;
      then p in rng(f) by A1,Th3;
      hence thesis by A5;
    end;
    0+1 <= len f by A1,NAT_1:13;
    then
A7: len f in dom f by FINSEQ_3:25;
    Suc(f) = f.(len f) by A1,Def2;
    then Suc(f) in rng(f) by A7,FUNCT_1:3;
    hence J,v |= Suc(f) by A5;
  end;
end;
