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:];
reserve a for Element of A;

theorem Th58:
  still_not-bound_in (f^g) = still_not-bound_in f \/ still_not-bound_in g
proof
  thus still_not-bound_in (f^g) c= still_not-bound_in f \/ still_not-bound_in g
  proof
    let b be object;
    assume b in still_not-bound_in (f^g);
    then consider i,p such that
A1: i in dom (f^g) and
A2: p = (f^g).i & b in still_not-bound_in p by Def5;
A3: now
      given n being Nat such that
A4:   n in dom g and
A5:   i = len f + n;
      (f^g).i = g.n by A4,A5,FINSEQ_1:def 7;
      then
A6:   b in still_not-bound_in g by A2,A4,Def5;
      still_not-bound_in g c= still_not-bound_in f \/ still_not-bound_in
      g by XBOOLE_1:7;
      hence thesis by A6;
    end;
    now
      assume
A7:   i in dom f;
      then (f^g).i = f.i by FINSEQ_1:def 7;
      then
A8:   b in still_not-bound_in f by A2,A7,Def5;
      still_not-bound_in f c= still_not-bound_in f \/ still_not-bound_in g
      by XBOOLE_1:7;
      hence thesis by A8;
    end;
    hence thesis by A1,A3,FINSEQ_1:25;
  end;
  thus still_not-bound_in f \/ still_not-bound_in g c= still_not-bound_in (f^g
  )
  proof
    let b be object such that
A9: b in still_not-bound_in f \/ still_not-bound_in g;
A10: now
      assume b in still_not-bound_in g;
      then consider i,p such that
A11:  i in dom g & p = g.i and
A12:  b in still_not-bound_in p by Def5;
      len f + i in dom (f^g) & p = (f^g).(len f + i) by A11,FINSEQ_1:28,def 7;
      hence thesis by A12,Def5;
    end;
    now
      assume b in still_not-bound_in f;
      then consider i,p such that
A13:  i in dom f and
A14:  p = f.i and
A15:  b in still_not-bound_in p by Def5;
      dom f c= dom (f^g) & p = (f^g).i by A13,A14,FINSEQ_1:26,def 7;
      hence thesis by A13,A15,Def5;
    end;
    hence thesis by A9,A10,XBOOLE_0:def 3;
  end;
end;
