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);

theorem Th3:
  len f > 0 implies f = Ant(f)^<*Suc(f)*> & rng f = rng Ant(f) \/ { Suc(f)}
proof
  assume
A1: len f > 0;
  then
A2: len f = len Ant(f)+1 by Th2;
A3: dom f = Seg len f by FINSEQ_1:def 3;
A4: now
    let j be Nat such that
A5: j in dom f;
A6: 1 <= j by A3,A5,FINSEQ_1:1;
A7: now
      assume j <= len Ant(f);
      then
A8:   j in dom Ant(f) by A6,FINSEQ_3:25;
      Ant(f) = f|Seg len Ant(f) by A2,Def1;
      then Ant(f) = f|dom Ant(f) by FINSEQ_1:def 3;
      then f.j = (Ant(f)).j by A8,FUNCT_1:49;
      hence f.j = (Ant(f)^<*Suc(f)*>).j by A8,FINSEQ_1:def 7;
    end;
A9: now
      1 in Seg 1 by FINSEQ_1:1;
      then
A10:  1 in dom <*Suc(f)*> by FINSEQ_1:38;
      assume
A11:  j = len Ant(f)+1;
      then j = len f by A1,Th2;
      then f.j = Suc(f) by A1,Def2;
      then f.j = <*Suc(f)*>.1;
      hence f.j = (Ant(f)^<*Suc(f)*>).j by A11,A10,FINSEQ_1:def 7;
    end;
    j <= len Ant(f)+1 by A2,A3,A5,FINSEQ_1:1;
    hence f.j = (Ant(f)^<*Suc(f)*>).j by A7,A9,NAT_1:8;
  end;
  len f = len Ant(f) + len <*Suc(f)*> by A2,FINSEQ_1:39;
  then
A12: len f = len (Ant(f)^<*Suc(f)*>) by FINSEQ_1:22;
  then f = (Ant(f)^<*Suc(f)*>) by A4,FINSEQ_2:9;
  then rng f = rng Ant(f) \/ rng <*Suc(f)*> by FINSEQ_1:31;
  hence thesis by A12,A4,FINSEQ_1:38,FINSEQ_2:9;
end;
